I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.

If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.

Basics

I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.

Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.

From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.

This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.

Intermediate

At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.

This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.

Advanced

Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.

Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.

Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.

Imagine you have a dataset without labels, but you want to solve a supervised problem with it, so you're going to try to collect labels. Let's say they are pictures of dogs and cats and you want to create labels to classify them.

One thing you could do is the following process:

1. Get a picture from your dataset.
   
2. Show it to a human and ask if it's a cat or a dog.
   
3. If the person says it's a cat or dog, mark it as a cat or dog.
   
4. Repeat.
   
   (I'm ignoring problems like pictures that are difficult to classify or lazy or adversarial humans giving you noisy labels)
   
   That's one way to do it, but is it the most efficient way? Imagine all your pictures are from only 10 cats and 10 dogs. Suppose they are sorted by individual. When you label the first picture, you get some information about the problem of classifying cats and dogs. When you label another picture of the same cat, you gain less information. When you label the 1238th picture from the same cat you probably get almost no information at all. So, to optimize your time, you should probably label pictures from other individuals before you get to the 1238th picture.
   
   How do you learn to do that in a principled way?
   
   Active Learning is a task where instead of first labeling the data and then learning a model, you do both simultaneously, and at each step you have a way to ask the model which next example should you manually classify for it to learn the most. You can than stop when you're already satisfied with the results.
   
   You could think of it as a reinforcement learning task where the reward is how much you'll learn for each label you acquire.
   
   The reason why, as a Bayesian, I like active learning, is the fact that there's a very old literature in Bayesian inference about what they call Experiment Design.
   
   Experiment Design is the following problem: suppose I have a physical model about some physical system, and I want to do some measurements to obtain information about the models parameters. Those measurements typically have control variables that I must set, right? What are the settings for those controls that, if I take measurements on that settings, will give the most information about the parameters?
   
   As an example: suppose I have an electric motor, and I know that its angular speed depends only on the electric tension applied on the terminals. And I happen to have a good model for it: it grows linearly up to a given value, and then it becomes constant. This model has two parameters: the slope of the linear growth and the point where it becomes constant. The first looks easy to determine, the second is a lot more difficult. I'm going to measure the angular speed at a bunch of different voltages to determine those two parameters. The set of voltages I'm going to measure at is my control variable. So, Experiment Design is a set of techniques to tell me what voltages I should measure at to learn the most about the value of the parameters.
   
   I could do Bayesian Iterated Experiment Design. I have an initial prior distribution over the parameters, and use it to find the best voltage to measure at. I then use the measured angular velocity to update my distribution over the parameters, and use this new distribution to determine the next voltage to measure at, and so on.
   
   How do I determine the next voltage to measure at? I have to have a loss function somehow. One possible loss function is the expected value of how much the accuracy of my physical model will increase if I measure the angular velocity at a voltage V, and use it as a new point to adjust the model. Another possible loss function is how much I expect the entropy of my distribution over parameters to decrease after measuring at V (the conditional mutual information between the parameters and the measurement at V).
   
   Active Learning is just iterated experiment design for building datasets. The control variable is which example to label next and the loss function is the negative expected increase in the performance of the model.
   
   So, now your procedure could be:
   
   
5. Start with:
   
   • a model to predict if the picture is a cat or a dog. It's probably a shit model.
     
   • a dataset of unlabeled pictures
     
   • a function that takes your model and a new unlabeled example and spits an expected reward if you label this example
     
6. Do:
   
   1. For each example in your current unlabeled set, calculate the reward
      
   2. Choose the example that have the biggest reward and label it.
      
   3. Continue until you're happy with the performance.
      
7. ????
   
8. Profit
   
   Or you could be a lot more clever than that and use proper reinforcement learning algorithms. Or you could be even more clever and use "model-independent" (not really...) rewards like the mutual information, so that you don't over-optimize the resulting data set for a single choice of model.
   
   I bet you have a lot of concerns about how to do this properly, how to avoid overfitting, how to have a proper train-validation-holdout sets for cross validation, etc, etc, and those are all valid concerns for which there are answers. But this is the gist of the procedure.
   
   You could do Active Learning and iterated experiment design without ever hearing about bayesian inference. It's just that those problems are natural to frame if you use bayesian inference and information theory.
   
   About the jargon, there's no way to understand it without studying bayesian inference and machine learning in this bayesian perspective. I suggest a few books:
   
   

• Information Theory, Inference, and Learning Algorithms, David Mackay - for which you can get a pdf or epub for free at this link.
  
  Is a pretty good introduction to Information Theory and bayesian inference, and how it relates to machine learning. The Machine Learning part might be too introductory if already know and use ML.
  
  
• Bayesian Reasoning and Machine Learning by David Barber - for which you can also get a free pdf here
  
  Some people don't like this book, and I can see why, but if you want to learn how bayesians think about ML, it is the most comprehensive book I think.
  
  
• Probability Theory, the Logic of Science by E. T. Jaynes. Free pdf of the first few chapters here.
  
  More of a philosophical book. This is a good book to understand what bayesians find so awesome about bayesian inference, and how they think about problems. It's not a book to take too seriously though. Jaynes was a very idiosyncratic thinker and the tone of some of the later chapters is very argumentative and defensive. Some would even say borderline crackpot. Read the chapter about plausible reasoning, and if that doesn't make you say "Oh, that's kind of interesting...", than nevermind. You'll never be convinced of this bayesian crap.
  
  

Sure! I'll just assume knowledge of the more common stuff like OPS. I'll try to break it into learning resources v. interesting work to be read. Think my suggestions to OP might be structured a bit differently. I'll try to keep it moderately short.

​

Learning

The Book: Playing the Percentages in Baseball set the foundation for a lot of stuff seen today. Win expectancy, lineup optimization, "clutch" hitting, matchups, etc. A lot of it is common knowledge today, but probably because of this work. It's great to see them work through it.

This is a bit of a glossary to many of the more important stats, with links for further reading.

As well, not quite the same, but Analyzing Baseball Data With R is also a great introduction to learning R, which is probably preferable to Python for a lot of baseball-specific work (not to make a general statement on the two, at all).

​

Reading

A lot of good work is, somewhat annoyingly, scattered through the internet on blogs. I don't have time to dig up too much right now but I'll shamelessly plug some work a couple of friends did a few years ago that was rather successful. These are mostly just examples of the what these projects tend to look like.

• The Value of Draft Picks
  
• Projecting Prospects' Hitting Primes
  
• xxFIP p1 p2 p3
  
  Much of the more current work will probably be found on FanGraphs' community submissions section, which I honestly haven't up with recently. I imagine a lot of focus is on using all the new Statcast data.
  
  There's also the MIT Sloan Sports Analytics Conference, where a lot of really cool work comes from. The awesome part about Sloan is that there seems to be a strong emphasis on sharing; I looked for the data/code for two papers I was interested in and ended up getting it for three! My favourite work might be (batter|pitcher)2vec. This is more machine-learning oriented, which I think is a good direction.
  
  ​
  
  That's all I have time for rn, hope that helps!

I'm usually pretty optimistic for people when it comes to posts asking about "how do I get started in sabermetrics" because I was in that position once as well, and it's worked out okay for me, but I want to be a bit more realistic, because I think there is a big red flag that you should recognize in yourself in respect to this.

There are a couple ways to get jobs in fields that require sabermetrics, but you should be aware: there are very few, they are highly competitive, and they require a good amount of work.

The traditional progression for doing sabermetric work is usually something like:

Stage|Level of Sabermetric Experience|Work you're qualified to do|
--:|:--|:--|
1|You look up stats online to form arguments about baseball|Personal blogging, entry-level analytics writing (FanSided, SBN, other sites)|
2|You put stats into a spreadsheet to visualize data or calculate something new to form an argument about baseball|Personal blogging, entry-level analytics writing (FanSided, SBN, other sites), heavier stuff if you're very lucky and a good writer (bigger sites like FanGraphs, Baseball Prospectus), general baseball coverage that isn’t heavily analytical|
3|You use code with baseball stats to visualize data or calculate something new to form an argument about baseball|Heavier analytics writing (SBN, FanGraphs, Baseball Prospectus, The Athletic), entry-level baseball operations work|
4|You use code to create your own models, predictions, and projections about baseball.|Extremely heavy analytics writing, baseball operations/team analytics work|

From your post, it sounds like you're somewhere between #1 and #2 right now. However: "after trying [coding] out I did not like it." You have a very large barrier keeping you from making the jump to stage 3.

If you actually want to go into a sabermetric field as a career, you need to know how to code. Not with Javascript, mind you, but other languages (Python, R, SQL, etc.). I would advise that you try out Python or R (Analyzing Baseball Data with R is an excellent introduction and gives you a lot of practical skills) and see if those really suck you in - and believe me, they need to suck you in. If you really don't like it, don't force yourself to do it and find some other career path, because you won't be able to succeed if you can't enjoy the work that you do.

FanSided has very low barriers of entry and the compensation reflects that - you cannot make a career out of blogging for FanSided. Even if you get to where I am (stage 4), if you're lucky, you might land a contributing position at a site that pays decently for part-time work. There are extremely few people who are somewhere between #3 and #4 who can make a full-time living off of baseball work, and they do it because they like what they do - if you don't like coding and working with baseball data in that environment, you're not going to be able to beat out everybody else who's trying to get there.

Let's say that you work your rear end off, you get to stage three or stage four. What options are available to you? There's maybe a handful of people who work in the "public" sector - that is, writing for websites like FanGraphs, Baseball Prospectus, The Athletic - who make enough money to make sabermetrics their full-time job. It will take a hail fucking mary to land one of those jobs, regardless of how talented you are, and you'll basically need to work double-duty on both sabermetrics and whatever your main hustle is until one of those positions opens up, and even then, you're not guaranteed anything.

You could also work for a team! There are far more positions available, they pay better, you have more data to work with, better job security - this sounds great, right? Problem is, the market cap for analysts are at about 20 per team, so there's something like 600 analyst positions that could be available in the future (I can't promise that the MLB will ever have 600 analysts total at any given time, but that's an upper estimate). And almost half of those are already full! There's not a whole lot of brain drain from the industry, so it is still extremely hard to break in and you're still going to be competing with the absolute best people in the industry. You will have to love to code and do this work because everybody you're competing with already does, and everybody else is willing to work twice as hard for it.

My advice to you is this: try out R or Python with baseball data. See if it's enough to get you addicted. See if it starts to occupy every ounce of free time you have, and you feel comfortable with it, and you're willing to put yourself out there and advertise your own work. I'm a full time student and basically every ounce of my free time is put towards working with this stuff, like it's a second full-time job for the past three years, and I'm still a bit of a ways away from making a living off of this. If you can't learn to love it, your time and energy are best spent elsewhere.

I don't, but I'm in the minority of the field. It definitely required a lot of catch-up in my first couple years. If you want to try and break in I can make some suggestions for self-teaching.

Linear Algebra is the backbone of all numerical modeling. I can make two suggestions to start with:

• I was very impressed with Jim Hefferon's book. It's part of an open courseware project so is available for free here (along with full solutions) but for $13 used I'd rather just have the book.
  
  
• The Gilbert Strang course on MIT Open Courseware is very good as well. I didn't like his book as well, but the video lectures are excellent as supplemental material for when I had questions from Hefferon.
  
  As for the actual FEA/CFD implementations:
  
  
• Numerical Heat Transfer and Fluid Flow ($22, used) seems to the standard reference for fluid flow. I'm relatively new to CFD so can't comment on it, but it seems to pop up constantly in any discussion of models or development.
  
  
• Finite Element Procedures, ($28, used) and the associated Open Courseware site. The solid mechanics (FEA) is very well done, again, haven't looked much at the fluids side.
  
  
• 12 steps to Navier Stokes. If you're interested in Fluids, start here. It's an excellent introduction and you can have a basic 2D Navier Stokes solver implemented in 48 hours.
  
  Note that none of these will actually teach you the the software side, but most commercial packages have very good tutorials available. These all teach the math behind what the solver is doing. You don't need to be an expert in it but should have a basic idea of what is going on.
  
  Also, OpenFoam is a surprisingly good open source CFD package with a strong community. I'd try and use it to supplement your existing work if possible, which will give you experience and make future positions easier. Play with this while you're learning the theory, don't approach it as "read books for two years, then try and run a simulation".

You're pretty good when it comes to linear vs. generalized linear models--and the comparison is the same regardless of whether you use mixed models or not. I don't agree at all with your "Part 3".

My favorite reference on the subject is Gelman & Hill. That book prefers to the terminology of "pooling", and considers models that have "no pooling", "complete pooling", or "partial pooling".

One of the introductory datasets is on Radon levels in houses in Minnesota. The response is the (log) Radon level, the main explanatory variable is the floor of the house the measurement was made: 0 for basement, 1 for first floor, and there's also a grouping variable for the county.

Radon comes out of the ground, so, in general, we expect (and see in the data) basement measurements to have higher Radon levels than ground floor measurements, and based on varied soil conditions, different overall levels in different counties.

We could fit 2 fixed effect linear models. Using R formula psuedocode, they are:

1. radon ~ floor
   
2. radon ~ floor + county (county as a fixed effect)
   
   The first is the "complete pooling" model. Everything is grouped together into one big pool. You estimate two coefficients. The intercept is the mean value for all the basement measurements, and your "slope", the floor coefficient, is the difference between the ground floor mean and the basement mean. This model completely ignores the differences between the counties.
   
   The second is the "no pooling" estimate, where each county is in it's own little pool by itself. If there are k counties, you estimate k + 1 coefficients: one intercept--the mean value in your reference county, one "slope", and k - 1 county adjustments which are the differences between the mean basement measurements in each county to the reference county.
   
   Neither of these models are great. The complete pooling model ignores any information conveyed by the county variable, which is wasteful. A big problem with the second model is that there's a lot of variation in how sure we are about individual counties. Some counties have a lot of measurements, and we feel pretty good about their levels, but some of the counties only have 2 or 3 data points (or even just 1). What we're doing in the "no pooling" model is taking the average of however many measurement there are in each county, even if there are only 2, and declaring that to be the radon level for that county. Maybe Lincoln County has only two measurements, and they both happen to be pretty high, say 1.5 to 2 standard deviations above the grand mean. Do you really think that this is good evidence that Lincoln County has exceptionally high Radon levels? Your model does, it's fitted line goes straight between the two Lincoln county points, 1.75 standard deviations above the grand mean. But maybe you're thinking "that could just be a fluke. Flipping a coin twice and seeing two heads doesn't mean the coin isn't fair, and having only two measurements from Lincoln County and they're both on the high side doesn't mean Radon levels there are twice the state average."
   
   Enter "partial pooling", aka mixed effects. We fit the model radon ~ floor + (1 | county). This means we'll keep the overall fixed effect for the floor difference, but we'll allow the intercept to vary with county as a random effect. We assume that the intercepts are normally distributed, with each county being a draw from that normal distribution. If a county is above the statewide mean and it has lots of data points, we're pretty confident that the county's Radon level is actually high, but if it's high and has only two data points, they won't have the weight to pull up the measurement. In this way, the random effects model is a lot like a Bayesian model, where our prior is the statewide distribution, and our data is each county.
   
   The only parameters that are actually estimated are the floor coefficient, and then the mean and SD of the county-level intercept. Thus, unlike the complete pooling model, the partial pooling model takes the county info into account, but it is far more parsimonious than the no pooling model. If we really care about the effects of each county, this may not be the best model for us to use. But, if we care about general county-level variation, and we just want to control pretty well for county effects, then this is a great model!
   
   Of course, random effects can be extended to more than just intercepts. We could fit models where the floor coefficient varies by county, etc.
   
   Hope this helps! I strongly recommend checking out Gelman and Hill.

Differential geometry track. I'll try to link to where a preview is available. Books are listed in something like an order of perceived difficulty. Check Amazon for reviews.

Calculus

Thompson, Calculus Made Easy. Probably a good first text, well suited for self-study but doesn't cover as much as the next two and the problems are generally much simpler. Legally available for free online.

Stewart, Calculus. Really common in college courses, a great book overall. I should also note that there is a "Stewart lite" called Calculus: Early Transcendentals, but you're better off with regular Stewart. Huh, it looks like there's a new series called Calculus: Concepts and Contexts which may be a good substitute for regular Stewart. Dunno.

Spivak, Calculus. More difficult, probably better than Stewart in some sense.

Linear Algebra

Poole, Linear Algebra. I haven't read this one but it has great reviews so I might as well include it.

Strang, Introduction to Linear Algebra. I think the Amazon reviews summarize how I feel about this book. Good for self-study.

Differential Geometry

Pressley, Elementary Differential Geometry. Great text covering curves and surfaces. Used this one in my undergrad course.

Do Carmo, Differential Geometry of Curves and Surfaces. Probably better left for a second course, but this one is the standard (for good reason).

Lee, Riemannian Manifolds: An Introduction to Curvature. After you've got a grasp on two and three dimensions, take a look at this. A great text on differential geometry on manifolds of arbitrary dimension.

------

Start with calculus, studying all the single-variable stuff. After that, you can either switch to linera algebra before doing multivariable calculus or do multivariable calculus before doing linear algebra. I'd probably stick with calculus. Pay attention to what you learn about vectors along the way. When you're ready, jump into differential geometry.

Hopefully someone can give you a good track for the other geometric subjects.

i have three categories of suggestions.

advanced calculus

these are essentially precursors to smooth manifold theory. you mention you have had calculus 3, but this is likely the modern multivariate calculus course.

• advanced calculus: a differential forms approach by harold edwards
  
  
• advanced calculus: a geometric view by james callahan
  
  
• vector calculus, linear algebra, and differential forms: a unified approach by john hubbard
  
  out of these, if you were to choose one, i think the callahan book is probably your best bet to pull from. it is the most modern, in both approach and notation. it is a perfect setup for smooth manifolds (however, all of these books fit that bill). hubbard's book is very similar, but i don't particularly like its notation. however, it has some unique features and does attempt to unify the concepts, which is a nice approach. edwards book is just fantastic, albeit a bit nonstandard. at a minimum, i recommend reading the first three chapters and then the latter chapters and appendices, in particular chapter 8 on applications. the first three chapters cover the core material, where chapters 4-6 then go on to solidify the concepts presented in the first three chapters a bit more rigorously.
  
  smooth manifolds
  
  
• an introduction to manifolds by loring tu
  
  
• introduction to smooth manifolds by john m. lee
  
  
• manifolds and differential geometry by jeffrey m. lee
  
  
• first steps in differential geometry: riemannian, contact, sympletic by andrew mcinerney
  
  out of these books, i only have explicit experience with the first two. i learned the material in graduate school from john m. lee's book, which i later solidifed by reading tu's book. tu's book actually covers the same core material as lee's book, but what makes it more approachable is that it doesn't emphasize, and thus doesn't require a lot of background in, the topological aspects of manifolds. it also does a better job of showing examples and techniques, and is better written in general than john m. lee's book. although, john m. lee's book is rather good.
  
  so out of these, i would no doubt choose tu's book. i mention the latter two only to mention them because i know about them. i don't have any experience with them.
  
  conceptual books
  
  these books should be helpful as side notes to this material.
  
  
• div, grad, curl are dead by william burke [pdf]
  
  
• geometrical vectors by gabriel weinreich
  
  
• about vectors by banesh hoffmann
  
  i highly recommend all of these because they're all rather short and easy reads. the first two get at the visual concepts and intuition behind vectors, covectors, etc. they are actually the only two out of all of these books (if i remember right) that even talk about and mention twisted forms.
  
  there are also a ton of books for physicists, applied differential geometry by william burke, gauge fields, knots and gravity by john baez and javier muniain (despite its title, it's very approachable), variational principles of mechanics by cornelius lanczos, etc. that would all help with understanding the intuition and applications of this material.
  
  conclusion
  
  if you're really wanting to get right to the smooth manifolds material, i would start with tu's book and then supplement as needed from the callahan and hubbard books to pick up things like the implicit and inverse function theorems. i highly recommend reading edwards' book regardless. if you're long-gaming it, then i'd probably start with callahan's book, then move to tu's book, all the while reading edwards' book. :)
  
  i have been out of graduate school for a few years now, leaving before finishing my ph.d. i am actually going back through callahan's book (didn't know about it at the time and/or it wasn't released) for fun and its solid expositions and approach. edwards' book remains one of my favorite books (not just math) to just pick up and read.

FWIW I had no fun with mathematics in school and didn't start studying it til I was in my thirties. I'm no genius, but I now teach the subject and still self-study it. You don't need any mysterious talent to get very competent at university-level maths, just to be interested enough in it to put the hours in.

Self-study is hard and frustrating. Be prepared for that. Reading one page can take a day. You can stare at a definition or theorem for hours and not understand it. Looking things up in multiple books can really help with that -- there are some good resources online as well. Also, some things just take a while to "cook" in the brain; keep at it. Take lots and lots of notes, preferably with pictures. Do plenty of exercises. When you're really stumped, post here.

I'll echo what others have said: add to Spivak a couple of other books so you can change it up. A book on group theory and one on linear algebra would be a nice combination -- maybe one on discrete maths, probability or something similar as well if that interests you. For group theory I think this book is fantastic, though it's expensive.

If you really want to make it through Spivak, make a plan. Break the book down into, say, 50-page chunks and make 50 pages your target for each week (I have no idea whether this is too ambitious for you -- try it and see). Track your progress. Celebrate when you hit milestones.

Good luck!

[EDIT: Also, be aware that maths books aren't really designed to be read like novels. Skim a chapter first looking for the highlights and general ideas, then drill into some of the details. Skip things that seem difficult and see if they become important later, then go back (with more motivation) etc.]

Entrepreneur Reading List

1. Disrupted: My Misadventure in the Start-Up Bubble
   
2. The Phoenix Project: A Novel about IT, DevOps, and Helping Your Business Win
   
3. The E-Myth Revisited: Why Most Small Businesses Don't Work and What to Do About It
   
4. The Art of the Start: The Time-Tested, Battle-Hardened Guide for Anyone Starting Anything
   
5. The Four Steps to the Epiphany: Successful Strategies for Products that Win
   
6. Permission Marketing: Turning Strangers into Friends and Friends into Customers
   
7. Ikigai
   
8. Reality Check: The Irreverent Guide to Outsmarting, Outmanaging, and Outmarketing Your Competition
   
9. Bootstrap: Lessons Learned Building a Successful Company from Scratch
   
10. The Marketing Gurus: Lessons from the Best Marketing Books of All Time
    
11. Content Rich: Writing Your Way to Wealth on the Web
    
12. The Web Startup Success Guide
    
13. The Best of Guerrilla Marketing: Guerrilla Marketing Remix
    
14. From Program to Product: Turning Your Code into a Saleable Product
    
15. This Little Program Went to Market: Create, Deploy, Distribute, Market, and Sell Software and More on the Internet at Little or No Cost to You
    
16. The Secrets of Consulting: A Guide to Giving and Getting Advice Successfully
    
17. The Innovator's Solution: Creating and Sustaining Successful Growth
    
18. Startups Open Sourced: Stories to Inspire and Educate
    
19. In Search of Stupidity: Over Twenty Years of High Tech Marketing Disasters
    
20. Do More Faster: TechStars Lessons to Accelerate Your Startup
    
21. Content Rules: How to Create Killer Blogs, Podcasts, Videos, Ebooks, Webinars (and More) That Engage Customers and Ignite Your Business
    
22. Maximum Achievement: Strategies and Skills That Will Unlock Your Hidden Powers to Succeed
    
23. Founders at Work: Stories of Startups' Early Days
    
24. Blue Ocean Strategy: How to Create Uncontested Market Space and Make Competition Irrelevant
    
25. Eric Sink on the Business of Software
    
26. Words that Sell: More than 6000 Entries to Help You Promote Your Products, Services, and Ideas
    
27. Anything You Want
    
28. Crossing the Chasm: Marketing and Selling High-Tech Products to Mainstream Customers
    
29. The Innovator's Dilemma: The Revolutionary Book that Will Change the Way You Do Business
    
30. Tao Te Ching
    
31. Philip & Alex's Guide to Web Publishing
    
32. The Tao of Programming
    
33. Zen and the Art of Motorcycle Maintenance: An Inquiry into Values
    
34. The Inmates Are Running the Asylum: Why High Tech Products Drive Us Crazy and How to Restore the Sanity
    
    

    Computer Science Grad School Reading List
    

    
    

35. All the Mathematics You Missed: But Need to Know for Graduate School
    
36. Introductory Linear Algebra: An Applied First Course
    
37. Introduction to Probability
    
38. The Structure of Scientific Revolutions
    
39. Science in Action: How to Follow Scientists and Engineers Through Society
    
40. Proofs and Refutations: The Logic of Mathematical Discovery
    
41. What Is This Thing Called Science?
    
42. The Art of Computer Programming
    
43. The Little Schemer
    
44. The Seasoned Schemer
    
45. Data Structures Using C and C++
    
46. Algorithms + Data Structures = Programs
    
47. Structure and Interpretation of Computer Programs
    
48. Concepts, Techniques, and Models of Computer Programming
    
49. How to Design Programs: An Introduction to Programming and Computing
    
50. A Science of Operations: Machines, Logic and the Invention of Programming
    
51. Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology
    
52. The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation
    
53. The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine
    
54. Computability: An Introduction to Recursive Function Theory
    
55. How To Solve It: A New Aspect of Mathematical Method
    
56. Types and Programming Languages
    
57. Computer Algebra and Symbolic Computation: Elementary Algorithms
    
58. Computer Algebra and Symbolic Computation: Mathematical Methods
    
59. Commonsense Reasoning
    
60. Using Language
    
61. Computer Vision
    
62. Alice's Adventures in Wonderland
    
63. Gödel, Escher, Bach: An Eternal Golden Braid
    
    

    Video Game Development Reading List
    

    
    

64. Game Programming Gems - 1 2 3 4 5 6 7
    
65. AI Game Programming Wisdom - 1 2 3 4
    
66. Making Games with Python and Pygame
    
67. Invent Your Own Computer Games With Python
    
68. Bit by Bit

Hi,
do you want to become a computer scientist or a programmer? That's the question you have to ask yourself. Just recently someone asked about some self-study courses in cs and I compiled a list of courses that focuses on the theoretical basics (roughly the first year of a bachelor class). Maybe it's helpful to you so I'm gonna copy&paste it here for you:



I think before you start you should ask yourself what you want to learn. If you're into programming or want to become a sysadmin you can learn everything you need without taking classes.

If you're interested in the theory of cs, here are a few starting points:

Introduction to Automata Theory, Languages, and Computation

The book you should buy

MIT: Introduction to Algorithms

The book you should buy


Computer Architecture<- The intro alone makes it worth watching!

The book you should buy

Linear Algebra

The book you should buy <-Only scratches on the surface but is a good starting point. Also it's extremely informal for a math book. The MIT-channel offers many more courses and are a great for autodidactic studying.

Everything I've posted requires no or only minimal previous education.
You should think of this as a starting point. Maybe you'll find lessons or books you'll prefer. That's fine! Make your own choices. If you've understood everything in these lessons, you just need to take a programming class (or just learn it by doing), a class on formal logic and some more advanced math classes and you will have developed a good understanding of the basics of cs. The materials I've posted roughly cover the first year of studying cs. I wish I could tell you were you can find some more math/logic books but I'm german and always used german books for math because they usually follow a more formal approach (which isn't necessarily a good thing).
I really recommend learning these thing BEFORE starting to learn the 'useful' parts of CS like sql,xml, design pattern etc.
Another great book that will broaden your understanding is this Bertrand Russell: Introduction to mathematical philosophy
If you've understood the theory, the rest will seam 'logical' and you'll know why some things are the way they are. Your working environment will keep changing and 20 years from now, we will be using different tools and different languages, but the theory won't change. If you've once made the effort to understand the basics, it will be a lot easier for you to switch to the next 'big thing' once you're required to do so.

One more thing: PLEASE, don't become one of those people who need to tell everyone how useless a university is and that they know everything they need just because they've been working with python for a year or two. Of course you won't need 95% of the basics unless you're planning on staying in academia and if you've worked instead of studying, you will have a head start, but if someone is proud of NOT having learned something, that always makes me want to leave this planet, you know...

EDIT: almost forgot about this: use Unix, use Unix, and I can't emphasize this enough: USE UNIX! Building your own linux from scratch is something every computerscientist should have done at least once in his life. It's the only way to really learn how a modern operating system works. Also try to avoid apple/microsoft products, since they're usually closed source and don't give you the chance to learn how they work.

Personally I make a distinction between scripting and programming that doesn't really exist but highlights the differences I guess. I consider myself to be scripting if I am connecting programs together by manipulating input and output data. There is lots of regular expression pain and trial-and-error involved in this and I have hated it since my first day of research when I had to write a perl script to extract the energies from thousands of gaussian runs. I appreciate it, but I despise it in equal measure. Programming I love, and I consider this to be implementing a solution to a physical problem in a stricter language and trying to optimise the solution. I've done a lot of this in fortran and java (I much prefer java after a steep learning curve from procedural to OOP). I love the initial math and understanding, the planning, the implementing and seeing the results. Debugging is as much of a pain as scripting, but I've found the more code I write the less stupid mistakes I make and I know what to look for given certain error messages. If I could just do scientific programming I would, but sadly that's not realistic. When you get to do it it's great though.

The maths for comp chem is very similar to the maths used by all the physical sciences and engineering. My go to reference is Arfken but there are others out there. The table of contents at least will give you a good idea of appropriate topics. Your university library will definitely have a selection of lower-level books with more detail that you can build from. I find for learning maths it's best to get every book available and decide which one suits you best. It can be very personal and when you find a book by someone who thinks about the concepts similarly to you it is so much easier.
For learning programming, there are usually tutorials online that will suffice. I have used O'Reilly books with good results. I'd recommend that you follow the tutorials as if you need all of the functionality, even when you know you won't. Otherwise you get holes in your knowledge that can be hard to close later on. It is good supplementary exercise to find a method in a comp chem book, then try to implement it (using google when you get stuck). My favourite algorithms book is Numerical Recipes - there are older fortran versions out there too. It contains a huge amount of detailed practical information and is geared directly at computational science. It has good explanations of math concepts too.

For the actual chemistry, I learned a lot from Jensen's book and Leach's book. I have heard good things about this one too, but I think it's more advanced. For Quantum, there is always Szabo & Ostlund which has code you can refer to, as well as Levine. I am slightly divorced from the QM side of things so I don't have many other recommendations in that area. For statistical mechanics it starts and ends with McQuarrie for me. I have not had to understand much of it in my career so far though. I can also recommend the Oxford Primers series. They're cheap and make solid introductions/refreshers. I saw in another comment you are interested potentially in enzymology. If so, you could try Warshel's book which has more code and implementation exercises but is as difficult as the man himself.

Jensen comes closest to a detailed, general introduction from the books I've spent time with. Maybe focus on that first. I could go on for pages and pages about how I'd approach learning if I was back at undergrad so feel free to ask if you have any more questions.



Out of curiosity, is it DLPOLY that's irritating you so much?

For math you're going to need to know calculus, differential equations (partial and ordinary), and linear algebra.

For calculus, you're going to start with learning about differentiating and limits and whatnot. Then you're going to learn about integrating and series. Series is going to seem a little useless at first, but make sure you don't just skim it, because it becomes very important for physics. Once you learn integration, and integration techniques, you're going to want to go learn multi-variable calculus and vector calculus. Personally, this was the hardest thing for me to learn and I still have problems with it.

While you're learning calculus you can do some lower level physics. I personally liked Halliday, Resnik, and Walker, but I've also heard Giancoli is good. These will give you the basic, idealized world physics understandings, and not too much calculus is involved. You will go through mechanics, electromagnetism, thermodynamics, and "modern physics". You're going to go through these subjects again, but don't skip this part of the process, as you will need the grounding for later.

So, now you have the first two years of a physics degree done, it's time for the big boy stuff (that is the thing that separates the physicists from the engineers). You could get a differential equations and linear algebra books, and I highly suggest you do, but you could skip that and learn it from a physics reference book. Boaz will teach you the linear and the diffe q's you will need to know, along with almost every other post-calculus class math concept you will need for physics. I've also heard that Arfken, Weber, and Harris is a good reference book, but I have personally never used it, and I dont' know if it teaches linear and diffe q's. These are pretty much must-haves though, as they go through things like fourier series and calculus of variations (and a lot of other techniques), which are extremely important to know for what is about to come to you in the next paragraph.

Now that you have a solid mathematical basis, you can get deeper into what you learned in Halliday, Resnik, and Walker, or Giancoli, or whatever you used to get you basis down. You're going to do mechanics, E&M, Thermodynamis/Statistical Analysis, and quantum mechanics again! (yippee). These books will go way deeper into theses subjects, and need a lot more rigorous math. They take that you already know the lower-division stuff for granted, so they don't really teach those all that much. They're tough, very tough. Obvioulsy there are other texts you can go to, but these are the one I am most familiar with.

A few notes. These are just the core classes, anybody going through a physics program will also do labs, research, programming, astro, chemistry, biology, engineering, advanced math, and/or a variety of different things to supplement their degree. There a very few physicists that I know who took the exact same route/class.

These books all have practice problems. Do them. You don't learn physics by reading, you learn by doing. You don't have to do every problem, but you should do a fair amount. This means the theory questions and the math heavy questions. Your theory means nothing without the math to back it up.

Lastly, physics is very demanding. In my experience, most physics students have to pretty much dedicate almost all their time to the craft. This is with instructors, ta's, and tutors helping us along the way. When I say all their time, I mean up until at least midnight (often later) studying/doing work. I commend you on wanting to self-teach yourself, but if you want to learn physics, get into a classroom at your local junior college and start there (I think you'll need a half year of calculus though before you can start doing physics). Some of the concepts are hard (very hard) to understand properly, and the internet stops being very useful very quickly. Having an expert to guide you helps a lot.

Good luck on your journey!

Some great stuff here.

However, these are MUST READS.

First, for a good introduction to numbers, read:
Innumeracy
http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Consequences-Vintage/dp/0679726012

It explains how numbers work very, very well, in a non-technical fashion.

Second, read,
The Structure of Scientific Revolutions by Thomas Kuhn
This excellent, excellent easy to read book is simply THE BEST EXPLANATION OF HOW SCIENCE WORKS.

Next, The Way Things Work by David Macaulay. It is not a 'science' book, per se, more of an engineering book, but it is brilliantly written and beautifully illustrated.

Then, dive into Asimov. "Please Explain" is fantastic. Though dated, so is his Guide to Science.


The great thing about these books is that they are all very short and aimed at people who are not technically educated. From there I am sure you will be able to start conquering more material.

Honestly, Innumeracy and The Structure of Scientific Revolutions, alone, will fundamentally change the way you look at absolutely everything around you. Genuinely eye opening.

This may vary by school, but it's been my experience that there aren't a lot of classes explicitly labeled as "artificial intelligence" (especially at the undergraduate level). However, AI is a very broad and interdisciplinary field so one thing I would recommend is that you take courses from fields that form the foundation of AI. (Math, Statistics, Computer Science, Psychology, Philosophy, Neurobiology, etc.)

In addition, take advantage of the resources you can find online! Self-study the topics you're interested in and try to get some hands on experience if possible: read blogs, read papers , browse subreddits, program a game-playing AI, etc.

Given that you're specifically interested in reasoning:

• (From the sidebar) AITopics has a page on reasoning with some recommendations on where to start.
  
  
• I'm not an expert in this area but from what I've been exposed to I believe many of the state-of-the-art approaches to reasoning rely on bayesian statistics so I would look into learning more about it. I've heard good things about this book, the author also has some lectures available on youtube
  
  
• From what I understand, whether or not we should look to the human mind for inspiration in AI reasoning is a pretty controversial topic. However you may find it interesting, and taking a brief survey of the psychology of reasoning may be a good way to understand the types of problems involved in AI reasoning, if you aren't very familiar with the topic.
  
  
  *As a disclaimer: I'm fairly new to this field of study myself. What I've shared with you is my best understanding, but given my lack of experience it may not be completely accurate. (Anyone, please feel free to correct me if I'm mistaken on any of these points)

• OpenIntro is a free basic statistics book including R labs on their website (it may be too basic for you, idk).
  
• Norm Matloff has a free book on statistics here. Even though it's intended for computer scientists, it's contents are really universal and basic (and it uses R too).
  
• Statistical inference by Casella and Berger is considered a "classic". It's heavy on theory though.
  
• All of Statistics by Larry Wasserman is very concise but offers a very good overview (all theory).
  
• See this post or this one for a compilation of freely available textbooks on statistics.
  
  As you wish to get into applied statistics (i.e. actually analyzing data), you'll need software. I'd strongly recommend learning and using R because it's completely free and incredibly powerful.
  
  Here are some resources for learning statistics using R:
  
  
• Statistics: An Introduction Using R by Michael Crawley introduces basic statistical methodology and its application with R. An alternative would be Introductory Statistics with R by Peter Dalgaard.
  
• http://www.r-statistics.com/2009/10/free-statistics-e-books-for-download/
  
• http://blog.revolutionanalytics.com/2011/11/three-free-books-on-r-for-statistics.html
  
• http://www.r-project.org/doc/bib/R-books.html
  
  Then, these websites provide very valuable resources for doing statistics with R:
  
  
• http://zoonek2.free.fr/UNIX/48_R/all.html
  
• http://www.cookbook-r.com/
  
• http://www.statmethods.net/
  
  Hope that helps.
  

Here is an actual blog post that conveys the width of the text box better. Here is a Tufte-inspired LaTeX package that is nice for writing papers and displaying side-notes; it is not necessary for now but will be useful later on. To use it, create a tex file and type the following:

\documentclass{article}
\usepackage{tufte-latex}

\begin{document}
blah blah blah
\end{document}

But don't worry about it too much; for now, just look at the Sample handout to get a sense for what good design looks like.

I mention AoPS because they have good problem-solving books and will deepen your understanding of the material, plus there is an emphasis on proof-writing when solving USA(J)MO and harder problems. Their community and resources tabs have many useful things, including a LaTeX tutorial.

Free intro to proofs books/course notes are a google search away and videos on youtube/etc too. You can also get a free library membership as a community member at a nearby university to check out books. Consider Aluffi's notes, Chartrand, Smith et al, etc.

You can also look into Analysis with intro to proof, a student-friendly approach to abstract algebra, an illustrated theory of numbers, visual group theory, and visual complex analysis to get some motivation. It is difficult to learn math on your own, but it is fulfilling once you get it. Read a proof, try to break it down into your own words, then connect it with what you already know.

Feel free to PM me v2 of your proof :)

Hello!

It's been a while since I last suggested a resource for calculus - so far, I've been finding the following two books extremely helpful and thought it would be good to share them:

1. The Calculus Lifesaver
   http://www.amazon.com/The-Calculus-Lifesaver-Tools-Princeton/dp/0691130884/ref=sr_1_1?ie=UTF8&qid=1398747841&sr=8-1&keywords=the+calculus+lifesaver
   
   I have mostly been using this as my main source of calculus lessons. You can find the corresponding lectures on youtube - the ones on his site do not work for whatever reason. The material is quite good, but still slightly challenging to ingest (though still much better than other courses out there!).
   
   
2. How to Ace Calculus: The Street-Wise Guide
   
   When I first saw this book, I thought it was going to be dumb, but I've been finding it extremely helpful. This is the book I'm using to understand some of the concepts in Calculus that are taken for granted (but that I need explained more in detail). It actually is somewhat entertaining while doing an excellent job of teaching calculus.
   
   The previous website I recommended to you is quite good at giving you an alternative perspective of calculus, but is not enough to actually teach you how to derive or integrate functions on your own. Hope your journey in math is going well!
   
   

I'm in a similar boat as you. I'm a biologist by trade, but want to delve deeper into statistical analysis with R programming to add a new skill to my career. I'm also a huge baseball fan, especially love it for the stats.

A friend of mine gave me this book for a birthday gift and I've been working way my through it, albeit very slowly. So far (I'm only at Chapter 3), it's been easy to follow and a nice to guide through R. I'd suggest it.

The edx course, that /u/sin7 suggested sounds interesting as well.

> To be honest, I do still think that step 2 is a bit suspect. The inverse of [;AA;]is [;(AA)^{-1};] . Saying that it's [;A^{-1}A^{-1};] seems to be skipping over something.

I realized how right you are when you say this after I reread the chapter on Inverse Matrices in my book. I am using Introduction to Linear Algebra by Gilbert Strang btw. I'm following his course on MIT OCW.

The book saids: If [;A;] and [;B;] are invertible then so is [;AB;]. The inverse of a product [;AB;] is [;(AB)^{-1}=B^{-1}A^{-1};].

So, before I went through with step two, I would have to have proved that [;A;] is indeed invertible.

>Their proof is basically complete. You could add the step from A2B to (AA)B which is equivalent to A(AB) due to the associativity from matrix multiplication and then refer to the definition of invertibility to say that A(AB) = I means that AB is the inverse of A. So you can make it a bit more wordy (and perhaps more clear), but the basic ingredients are all there.

I will write up the new proof right here, in its entirety. Please let me know what you think and what I need to fix and/or add.

Theorem: if [;B;] is the inverse of [;A^2;], then [;AB;] is the inverse of A.

Proof: Assume [;B;] is the inverse of [;A^2;]

1. Since [;B;] is the inverse of [;A^2;], we can say that [;A^2B=I;]
   
   
2. We can write [;A^2B=I;] as [;(AA)B=I;]
   
   
3. We can rewrite [;(AA)B=I;] as [;A(AB)=I;] because of the associative property of matrix multiplication.
   
   
4. Therefore, by the definition of matrix invertibility, since [;A(AB)=I;], [;AB;]is indeed the inverse of [;A;].
   
   Q.E.D.
   
   Do I have to include anything about the proof being correct for a right-inverse and a left-inverse?
   
   > That's a great initiative! Probably means you're already ahead of the curve. Even if you get a step (arguably) wrong, you're still practicing with writing up proofs, which is good. Your write-up looks good to me, except for the questionability of step 2. In step 3 (and possibly others) you might also want to mention what you are doing exactly. You say "therefore", but it might be slightly clearer if you explicitly mention that you're using your assumption. You can also number everything (including the assumption), and then put "combining statement 0 and 2" to the right (where you can also go into a bit more detail: e.g. "using associativity of multiplication on statement 4").
   
   I haven't began my studies at university yet, but I sure am glad that I exposed myself to proofs before taking an actual discrete math class. I think that very few people get exposed to proof writing in the U.S. public school system. I've completed all of the Khan Academy math courses, and the MIT OCW Math for CS course is still very difficult. I basically want to develop a very strong foundation in proof writing, and all the core courses I will take as a CS major now, and then I will hopefully have an easier time with my schoolwork once I begin in the fall. Hopefully this prior knowledge will keep my GPA high too. I really appreciate all the constructive criticism about my proof. I will try to make them as detailed as possible from now on.
   

I took both precalc and calc 1 back to back (and we used Stewart's calc book for calc 1-3). To be honest, concepts like limits and continuity aren't even covered in precalculus, so it isn't like you've missed something huge by skipping precalc. My precalc class was a lot of higher level college algebra review and then lots and lots and lots of trig.

I honestly don't see how you'd need much else aside from PatricJMT and lots of example problems. It may be worthwhile for you to pick up "The Calculus Lifesaver" by Adrian Banner. It's a really great book that breaks down the calc 1 concepts pretty well. Master limits because soon you'll move onto differentiation and then everything builds from that.

Precalc was my trig review that I was thankful for when I got to calc 2, however, so if you find yourself needing calculus 2, please review as much trig as you can. If you need some resources for trig review, PM me. I tutored college algebra, precalc, and calc for 3 years.

Good luck!

I realized as I was writing this reply, I'm not sure if you're interested in a general linear algebra reference material recommendation, or more of a computer graphics math recommendation. My reply is all about general linear algebra, but I don't think matrix decompositions or eigensolvers are used in real-time computer graphics (but what do I know lol), so probably just focusing on the transformations chapter in Mathematics for 3D Game Programming and Computer Graphics would be good. If it feels like you're just memorizing stuff, I think that's normal, but keep rereading the material and do examples by hand! If you really understand how projection matrices work, then the transformations should make more sense and seem less like magic.

I took Linear Algebra last semester and we used http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Edition/dp/0980232716, I would highly recommend it. Along with that book, I would recommend watching these video lectures, http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/, given by the author of the book. I've never watched MIT's video lectures until I watched these in preparation for an interview, because I always thought they would be dumb, but they're actually really great! I will say that I used the pause button furiously because the lectures are very dense and I had to think about what he was saying!

In my opinion, the most important topics to focus on would be the definition of a vector space, the four fundamental subspaces, how the four fundamental subspaces relate to the fundamental theorem of linear algebra, all the matrix decompositions in that book, pivot variables and special solutions...I just realized I'm basically listing all of the chapters in the book, but I really do think they are all very important! The one thing you might not want to focus on is the chapter on incidence matrices. However, in my class, we went over PageRank in detail and I think it was very interesting!

Anti-disclaimer: I do have personal experience with all the below books.

I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.

It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.

If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.

I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .

Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.

Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.

You're English is great.

I'd like to reemphasize /u/Plaetean's great suggestion of learning the math. That's so important and will make your later career much easier. Khan Academy seems to go all through differential equations. All of the more advanced topics they have differential and integral calculus of the single variable, multivariable calculus, ordinary differential equations, and linear algebra are very useful in physics.

As to textbooks that cover that material I've heard Div, Grad, Curl for multivariable/vector calculus is good, as is Strang for linear algebra. Purcell an introductory E&M text also has an excellent discussion of the curl.

As for introductory physics I love Purcell's E&M. I'd recommend the third edition to you as although it uses SI units, which personally I dislike, it has far more problems than the second, and crucially has many solutions to them included, which makes it much better for self study. As for Mechanics there are a million possible textbooks, and online sources. I'll let someone else recommend that.

Books:

"Doing Bayesian Data Analysis" by Kruschke. The instruction is really clear and there are code examples, and a lot of the mainstays of NHST are given a Bayesian analogue, so that should have some relevance to you.

"Bayesian Data Analysis" by Gelman. This one is more rigorous (notice the obvious lack of puppies on the cover) but also very good.

Free stuff:

"Think Bayes" by our own resident Bayesian apostle, Allen Downey. This book introduces Bayesian stats from a computational perspective, meaning it lays out problems and solves them by writing Python code. Very easy to follow, free, and just a great resource.

Lecture: "Bayesian Statistics Made (As) Simple (As Possible)" again by Prof. Downey. He's a great teacher.

Both time series and regression are not strictly econometric methods per se, and there are a range of wonderful statistics textbooks that detail them. If you're looking for methods more closely aligned with econometrics (e.g. difference in difference, instrumental variables) then the recommendation for Angrist 'Mostly Harmless Econometrics' is a good one. Another oft-prescribed econometric text that goes beyond Angrist is Wooldridge 'Introductory Econometrics: A Modern Approach'.

For a very well considered and basic approach to statistics up to regression including an excellent treatment of probability theory and the basic assumptions of statistical methodology, Andy Field (and co's) books 'Discovering Statistics Using...' (SPSS/SAS/R) are excellent.

Two excellent all-rounders are Cohen and Cohen 'Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences' and Gelman and Hill 'Data Analysis Using Regression and Multilevel/Hierarchical Modelling' although I would suggest both are more advanced than I am guessing you need right now.

For time series I can recommend Rob Hyndman's book/s on forecasting (online copy freely available)

For longitudinal data analysis I really like Judith Singer's book 'Applied Longitudinal Data Analysis'.

It sounds however as if you're looking for a bit of a book to explain why you would want to use one method over another. In my experience I wanted to know this when I was just starting. It really comes down to your own research questions and the available data. For example I had to learn Longitudinal/fixed/random effects modelling because I had to do a project with a longitudinal survey. Only after I put it into practice (and completed my stats training) did I come to understand why the modelling I used was appropriate.

When you say everyday calculations I'm assuming you're talking about arithmetic, and if that's the case you're probably just better off using you're phone if it's too complex to do in you're head, though you may be interested in this book by Arthur Benjamin.

I'm majoring in math and electrical engineering so the math classes I take do help with my "everyday" calculations, but have never really helped me with anything non-technical. That said, the more math you know the more you can find it just about everywhere. I mean, you don't have to work at NASA to see the technical results of math, speech recognition applications like Siri or Ok Google on you're phone are insanely complex and far from a "solved" problem.

Definitely a ton of math in the medical field. MRIs and CT scanners use a lot of physics in combination with computational algorithms to create images, both of which require some pretty high level math. There's actually an example in one of my probability books that shows how important statistics can be in testing patients. It turns out that even if a test has a really high accuracy, if the condition is extremely rare there is a very high probability that a positive result for the test is a false positive. The book states that ~80% of doctors who were presented this question answered incorrectly.

I am a Strange Loop is about the theorem

Another book I recommend is David Foster-Wallace's Everything and More. It's a creative book all about infinity, which is a very important philosophical concept and relates to mind and machines, and even God. Infinity exists within all integers and within all points in space. Another thing the human mind can't empirically experience but yet bears axiomatic, essential reality. How does the big bang give rise to such ordered structure? Is math invented or discovered? Well, if math doesn't change across time and culture, then it has essential existence in reality itself, and thus is discovered, and is not a construct of the human mind. Again, how does logic come out of the big bang? How does such order and beauty emerge in a system of pure flux and chaos? In my view, logic itself presupposes the existence of God. A metaphysical analysis of reality seems to require that base reality is mind, and our ability to perceive and understand the world requires that base reality be the omniscient, omnipresent mind of God.

Anyway these books are both accessible. Maybe at some point you'd want to dive into Godel himself. It's best to listen to talks or read books about deep philosophical concepts first. Jay Dyer does a great job on that

https://www.youtube.com/watch?v=c-L9EOTsb1c&t=11s

If you are looking for something very calculus-based, this is the book I am familiar with that is most grounded in that. Though, you will need some serious probability knowledge, as well.

If you are looking for something somewhat less theoretical but still mathematical, I have to suggest my favorite. Statistics by William L. Hays is great. Look at the top couple of reviews on Amazon; they characterize it well. (And yes, the price is heavy for both books.... I think that is the cost of admission for such things. However, considering the comparable cost of much more vapid texts, it might be worth springing for it.)

Well the good news is that we have more resources available now than even 5 years ago. :) I'm in calc 1 right now, and was having trouble putting the pieces together into a whole that made sense. A few of my resources are classroom specific but many would be great for anyone not currently in a class.

Free:
www.khanacademy.org

free video lectures and practice problems on all manner of topics, starting with elementary algebra. You can start at the beginning and work your way through, or just start wherever.

http://ocw.mit.edu/index.htm

free online courses and lessons from MIT (!!) where you can watch lectures on a subject, do practice problems, etc. Use just for review or treat it like a course, it's up to you.

Cheap $$

http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606/ref=sr_1_1?ie=UTF8&qid=1331675661&sr=8-1

$10ish shipped for a book that translates calculus from math-professor to plain english, and is funny too.

http://www.amazon.com/Calculus-Lifesaver-Tools-Excel-Princeton/dp/0691130884/ref=pd_cp_b_1

$15 for a book that is 2-3x as thick as the previous one, a bit drier, but still very readable. And it covers Calc 1-3.

Beginner Resources: These are fantastic places to start for true beginners.

Introduction to Probability is an oldie but a goodie. This is a basic book about probability that is suited for the absolute beginner. Its written in an older style of english, but other than that it is a great place to start.

Bayes Rule is a really simple, really basic book that shows only the most basic ideas of bayesian stats. If you are completely unfamiliar with stats but have a basic understanding of probability, this book is pretty good.

A Modern Approach to Regression with R is a great first resource for someone who understands a little about probability but wants to learn more about the details of data analysis.

​

Advanced resources: These are comprehensive, quality, and what I used for a stats MS.

Statistical Inference by Casella and Berger (2nd ed) is a classic text on maximum likelihood, probability, sufficiency, large sample properties, etc. Its what I used for all of my graduate probability and inference classes. Its not really beginner friendly and sometimes goes into too much detail, but its a really high quality resource.

Bayesian Data Analysis (3rd ed) is a really nice resource/reference for bayesian analysis. It isn't a "cuddle up by a fire" type of book since it is really detailed, but almost any topic in bayesian analysis will be there. Although its not needed, a good grasp on topics in the first book will greatly enhance the reading experience.

I recommend starting by pointing people at Nathan Carter’s book Visual Group Theory (site, amzn), which does a great job providing a number of examples of interesting groups, and motivating understanding of quotient groups etc.

If they start with a stable of nice nice concrete group theory examples, then a lot of the drier and more abstract parts of a “definition, theorem, proof, corollary, definition, ...” style presentation afterward will be much less cryptic.

The basic examples for ring theory / field theory are a bit more familiar to most students, so I expect the rest of abstract algebra usually turns out okay, though throwing in more concrete weird examples of rings and fields would be nice as well.

It would be interesting to start an introductory field theory course with a detailed concrete study of a few specific finite fields, and e.g. vector spaces and linear transformations and polynomials etc. worked out for those fields. It would give a very different flavor to the course.

Honestly, if you're wanting an understanding of statistics, I'd recommend Statistics for Dummies. Don't be deceived by the title, you'll still have to do some real thinking on your own to grasp the ideas discussed. You might consider using textbooks or other online resources as secondary supports to your study.

I can also give you a basic breakdown of the topics you'd want to develop an understanding of in beginning to study statistics.

Descriptive Statistics

Descriptive statistics is all about just describing your sample. Major ideas in being able to describe the sample are measures of center (e.g., mean, median, mode), measures of variation (e.g., standard deviation, variance, range, interquartile range), and distributions (e.g., uniform, bell-curve/normally distributed, skewed left/right).

Inferential Statistics

There is a TON of stuff related to this. However, I would first recommend beginning with making sure you have some basic understanding of probability (e.g., events, independence, mutual exclusivity) and then study sampling distributions. Because anything you make an inference about will depending upon the measures in your sample, you need to have a sense of what kinds of samples are possible (and most likely) when you gather data to form one. One of the most fundamental ideas of inferential statistics is based upon these ideas, The Central Limit Theorem. You'll want to make sure you understand what it means before progressing to making inferences.

With that background, you'll be ready to start studying different inferences (e.g., independent/dependent sample t-tests). Again, there are a lot of different kinds of inference tests out there, but I think the most important thing to emphasize with them is the importance of their assumptions. Various technologies will do all of the number crunching for you, but you have to be the one to determine if you're violating any assumptions of the test, as well as interpret what the results mean.

As a whole, I would encourage you to focus on understanding the big ideas. There is a lot of computation involved with statistics, but thanks to modern technology, you don't have to get bogged down in it. As a whole, keep pushing towards understanding the ideas and not getting bogged down in the fine-grained details and processes first, and it will help you develop a firm grasp of much of the statistics out there.

I invite you to read a couple of books, both of which I really enjoyed.

One, Two, Three...Infinity by George Gamow (that link is almost certainly an act of piracy, but I doubt the author would mind because he dedicated his life to spreading knowledge), and

Everything and More by David Foster Wallace. That's a publication which is so recent that if you want to read it, you'll have to cough up money. Go ahead and do it: it's so interesting that it's worth the eight bucks for the e-book easily. Don't worry, it's not an affiliate link, I stand to gain nothing from your purchase.

Both of those books talk about the mind-blowing idea that there are multiplie levels of infinity, with some infinities being much bigger than other infinities. The state of the art in thinking about infinities is so brain-hurting that David Foster Wallace's book was published 40 years after George Gamow's book, and includes a relatively small number of concepts that weren't in the older book (which isn't to say that they're insignificant--they're ideas about infinity so by necessity they're huge). One of the things I liked about David Foster Wallace's book is that it actually has a formula for quantifying how much bigger a higher-level infinity is than a lower-level infinity. Nobody in Gamow's day had come up with anything like that yet, they were just waving their arms talking about "huge" and "huger".

Here are some books I'd recommend.

General Books

These are general books that are more focused on proving things per se. They'll use examples from basic set theory, geometry, and so on.

1. How to Prove It: A Structured Approach by Daniel Velleman
   
2. How to Solve It: A New Aspect of Mathematical Method by George Pólya
   
   Topical Books
   
   For learning topically, I'd suggest starting with a topic you're already familiar with or can become easily familiar with, and try to develop more rigor around it. For example, discrete math is a nice playground to learn about proving things because the topic is both deep and approachable by a beginning math student. Similarly, if you've taken AP or IB-level calculus then you'll get a lot of out a more rigorous treatment of calculus.
   
   

• An Invitation to Discrete Mathematics by Jiří Matoušek and Jaroslav Nešetřil
  
• Discrete Mathematics: Elementary and Beyond by László Lovász and Jaroslav Pelikan
  
• Proofs from THE BOOK by Martin Aigner and Günter Ziegler
  
• Calculus by Michael Spivak
  
  I have a special place in my hear for Spivak's Calculus, which I think is probably the best introduction out there to math-as-she-is-spoke. I used it for my first-year undergraduate calculus course and realized within the first week that the "math" I learned in high school — which I found tedious and rote — was not really math at all. The folks over at /r/calculusstudygroup are slowly working their way through it if you want to work alongside similarly motivated people.
  
  General Advice
  
  One way to get accustomed to "proof" is to go back to, say, your Algebra II course in high school. Let's take something I'm sure you've memorized inside and out like the quadratic formula. Can you prove it?
  
  I don't even mean derive it, necessarily. It's easy to check that the quadratic formula gives you two roots for the polynomial, but how do you know there aren't other roots? You're told that a quadratic polynomial has at most two distinct roots, a cubic polynomial has a most three, a quartic as most four, and perhaps even told that in general an n^(th) degree polynomial has at most n distinct roots.
  
  But how do you know? How do you know there's not a third root lurking out there somewhere?
  
  To answer this you'll have to develop a deeper understanding of what polynomials really are, how you can manipulate them, how different properties of polynomials are affected by those manipulations, and so on.
  
  Anyways, you can revisit pretty much any topic you want from high school and ask yourself, "But how do I really know?" That way rigor (and proofs) lie. :)

(Note: I wrote this elsewhere)

Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:

• function
  
• set
  
• recursion
  
• proposition
  
• proof
  
  When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
  
  I don't know of any good online introduction to discrete math, but here are a few book ideas.
  
  
• For a good textbook on discrete mathematics, check Schaum's Outline of Discrete Mathematics. It's also cheap.
  
• For a less textbook-ish overview written in an entertaining manner, Discrete Mathematics: Elementary and Beyond is my personal favourite.
  
• If you want to learn-by-doing, then The Haskell Road to Logic, Maths and Programming teaches you discrete math by having you manipulate discrete math concepts in the programming language Haskell. Here is a good free online Haskell tutorial.

Not long! For this purpose I highly highly recommend Richard McElreath's Statistical Rethinking (this one here). It's SO good. The math is exceptionally straightforward for someone familiar with regression, and it's huge on developing intuition. Bonus: he sets you up with all the tools you need to do your own analyses, and there are tons of examples that he works from a lot of different angles. He even does hierarchical regression.

It's an easy math book to read cover to cover by yourself, to be honest. He really holds your hand the whole way through.

Jesus, he should pay me to rep his book.

Sure! I have a lot of resources on this subject. Before I recommend it, let me very quickly explain why it is useful.

Bayes Rule basically means creating a new hypothesis or belief based on a novel event using prior hypothesis/data. So I am sure you can already see how useful it would be in medicine to think about. The Rule(or technically theorem) is in fact an entire field of statisitcs and basically is one of the core parts of probability theory.

Bayes Rule explains why you shouldn't trust sensitivity and specificity as much as you think. It would take too long to explain here but if you look up Bayes' Theorem on wikipedia one of the first examples is about how despite a drug having 99% sensitivity and specificity, even if a user tests positive for a drug, they are in fact more likely to not be taking the drug at all.

Ok, now book recommendations:

Basic: https://www.amazon.com/Bayes-Theorem-Examples-Introduction-Beginners-ebook/dp/B01LZ1T9IX/ref=sr_1_2?ie=UTF8&qid=1510402907&sr=8-2&keywords=bayesian+statistics

https://www.amazon.com/Bayes-Rule-Tutorial-Introduction-Bayesian/dp/0956372848/ref=sr_1_6?ie=UTF8&qid=1510402907&sr=8-6&keywords=bayesian+statistics

Intermediate/Advanced: Only read if you know calculus and linear algebra, otherwise not worth it. That said, these books are extremely good and are a thorough intro compared to the first ones.

https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/ref=sr_1_1?ie=UTF8&qid=1510402907&sr=8-1&keywords=bayesian+statistics

https://www.amazon.com/Introduction-Probability-Chapman-Statistical-Science/dp/1466575573/ref=sr_1_12?s=books&ie=UTF8&qid=1510403749&sr=1-12&keywords=probability

My pleasure :\^) It's hard to say what a local community college would have, since courses seem to vary a lot from school to school. The best thing you could find would probably be a class on something like "Set Theory" or "Mathematical Thinking" (those usually tend to touch on subjects like this without being pathologically rigorous), but a course in Discrete Math could do the trick, since you often talk about counting which leads naturally to countable vs uncountable sets. If you really want to learn the hardcore math, a course in Real Analysis is what you want. And if you don't know where to begin or are too busy, I can't recommend this book enough: http://www.amazon.com/Everything-More-Compact-History-Infinity/dp/0393339289. It's DFW so you know it's good ;)

I'm actually an undergrad studying Computer Science and Math but yes, I plan to end up a teacher after some other sort of career. Feel free to PM me if you have any more questions.

It's pretty basic stuff, but the first three chapters of this book was a game-changer for me

https://www.amazon.com/Introduction-Probability-2nd-Dimitri-Bertsekas/dp/188652923X

My mind was blown when I finally understood the connection between random variables and the "basic" probability theory with events and sample spaces. For me they had always been two seperate things.

The notation is also really nice.

Having solid fundamentals makes it much easier to study advanced topics, so I would start here.

There's also a great EDX course which is based on the book, but it's a complement and not a substitute. Get the book.

The important part of this question is what do you know? By saying you're looking to learn "a little more about econometrics," does that imply you've already taken an undergraduate economics course? I'll take this as a given if you've found /r/econometrics. So this is a bit of a look into what a first year PhD section of econometrics looks like.

My 1st year PhD track has used
-Casella & Berger for probability theory, understanding data generating processes, basic MLE, etc.

-Greene and Hayashi for Cross Sectional analysis.

-Enders and Hamilton for Time Series analysis.

These offer a more mathematical treatment of topics taught in say, Stock & Watson, or Woodridge's Introductory Econometrics. C&B will focus more on probability theory without bogging you down in measure theory, which will give you a working knowledge of probability theory required for 99% of applied problems. Hayashi or Greene will mostly cover what you see in an undergraduate class (especially Greene, which is a go to reference). Hayashi focuses a bit more on general method of moments, but I find its exposition better than Greene. And I honestly haven't looked at Enders or Hamilton yet, but they will cover forecasting, auto-regressive moving average problems, and how to solve them with econometrics.

It might also be useful to download and practice with either R, a statistical programming language, or Python with the numpy library. Python is a very general programming language that's easy to work with, and numpy turns it into a powerful mathematical and statistical work horse similar to Matlab.

That is a pretty big red flag. Most departments offer a statistics course for non-math majors, I've TA'd that course before, its not good for people who enjoy math. Make sure there is at the very least a calculus pre-req for the course, and you should take a probability course first anyways.

Probability and Statistics in general is such a great field, it would be really unfortunate if a bad class designed for psych majors turned you off from the subject. I would wait until you can take the right classes so you can at least see the material presented in the right way, if your curious what a course sequence should look like:

1. Calculus based probability (undergrad, math major prob)
   
   
2. Calculus based statistics (undergrad, math major stat)
   
   
3. Measure-theoretic probability (grad prob)
   
   
4. Statistical Theory (grad stat - here is where analysis and some really neat math merges with the stats, but undergrad stat does this to some degree as well.
   
   Here are links to textbooks in the same order:
   
   
5. http://www.amazon.com/First-Course-Probability-9th-Edition/dp/032179477X
   
   
6. http://www.amazon.com/Mathematical-Statistics-Analysis-Available-Enhanced/dp/0534399428
   
   
7. http://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372
   
   
8. http://www.amazon.com/Mathematical-Statistics-Selected-Topics-Edition/dp/0132306379
   
   You can find most of those in pdf format somewhere online. I'm not saying those are the best textbooks to use, but they should at least provide a guide so that you can be sure you taking the right courses.
   
   Edit: If your school offers a combined prob/stat course (usually offered for engineers) that has a calc 3 pre-req it would probably serve well as a compact introduction to the subject.
   

> don't think that there is a logical progression to approaching mathematics

Well, this might be true of the field as a whole, but def not true when it comes to learning basic undergrad level math after calc 1, as the OP asked about. There are optimized paths to gaining mathematical maturity and sufficient background knowledge to read papers and more advanced texts.

> Go to the mathematics section of a library, yank any book off the shelf, and go to town.

I would definitely NOT do this, unless you have a lot of time to kill. I would, based on recommendations, pick good texts on linear algebra and differential equations and focus on those. I mean focus because it is easy in mathematics to gloss over difficulties.

My recommendation, since you are self-studying, is to pick up Gil Strang's linear algebra book (go for an older edition) and look up his video lectures on linear algebra. That's a solid place to start. I'd say that course could be done, with hard work, in a summer. For a differential equations book, I'm not exactly sure. I would seek out something with some solid applications in it, like maybe this: http://amzn.com/0387978941

That is more than a summer's worth of work.

Sorry, agelobear, to be such a contrarian.

Disclaimer: I'm an engineer, not a mathematician, so take my advice with a grain of salt.

Early in my grad degree I wanted to master probability and improve my understanding of statistics. The books I used, and loved, are

DeGroot, Probability and Statistics

Rozanov, Probability Theory: A Concise Course

The first is organized very well, with ever increasing difficulty and a good number of solved problems. I also appreciate that as things start to get complicated, he also always bridges everything back to earlier concepts. The books also basically does everything Bayesian and Frequentist side by side, so you get a really good idea of the comparison and arbitraryness.

The second is a good cheap short book basically full of examples. It has just enough math flavor to be mathier, without proofing me to death.

Also, if you're really just jumping into the subject, I would recommend some pop culture math books too, e.g.,

Paulos, Innumeracy

Mlodinow, The Drunkards Walk

Have fun!

I think for a rigorous treatment of linear algebra you'd want something like Strang's class book:

http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Edition/dp/0980232716

For me, what was great about this book was that it approached linear algebra via practical applications, and those applications were more relevant to computer science than pure mathematics, or electrical engineering like you find in older books. It's more about modern applications of LA. It's great for after you've studied the topic at a basic level. It's a great synthesis of the material.

It's a little loose, so if you have some basic chops, it's fantastic.

It sounds like you want some kind of regression, especially to answer 2. In a GLM, you are not claiming that the data by itself has a Normal/Poisson/Negative Binomial/Binomial distribution, only that it has such a distribution when conditioned on a number of factors.

In a nutshell: you model the mean of the distribution as a linear combination of the inputs. Then you can read the weighting factors on each input to learn about the relationship.

In other words, it doesn't need to be that your data is Poisson or NB in order to do a Poisson or NB regression. It only has to be that the error, that is, the difference between the expected based on the mean function and the actual, follows such a distribution. In fact, there may be some simple transformations (like taking the log of the outcome) that lets you use a standard linear model, where you can reasonably assume that the error is Normal, even if the outcome is anything but.

If your variance is not dependent on any of your inputs, that's a great sign, since heteroskedasticity is a great annoyance when trying to do regressions.

If you have time, the modern classic in this area is http://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Models/dp/052168689X. It starts with a pretty gentle introduction to regression and works its way into the cutting edge by the end.

If you want to do statistics in a rigorous way you should start with calculus and linear algebra.

For calculus I recommend Paul's notes -> http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
They are really clearly written with good examples and provide good intuition.
As supplement go through 3blue1borwn Essence of calculus. I think it's an excellent resource for providing the right intuition.

For linear algebra - linear algebra - Linear algebra done right as already recommended. Additionally, again 3blue1brown series on linear algebra are top notch addition for providing visual intuition and understanding for what is going on and what it's all about.

Finally, for statistics - I would recommend starting with probability calculus - that way you'll be able to do mathematical statistics and will have a solid understanding of what is going on. Mathematical statistics with applications is self-contained with probability calculus included. https://www.amazon.com/Mathematical-Statistics-Applications-Dennis-Wackerly/dp/0495110817

Since you are still in college, why not take a statistics class? Perhaps it can count as an elective for your major. You might also want to consider a statistics minor if you really enjoy it. If these are not options, then how about asking the professor if you can sit in on the lectures?

It sounds like you will be able to grasp programming in R, may I suggest trying out SAS? This book by Ron Cody is a good introduction to statistics with SAS programming examples. It does not emphasize theory though. For theory, I would recommend Casella & Berger, many consider this book to be a foundation for statisticians and is usually taught at a grad level.

Good luck!

The best way to learn is take the class and find your deficiencies. Khan Academy is also great to get a base line of where you are. If you need help with calc. And precal, calculus lifesaver book is good.
lifesaver calculus amazon

>All I'm saying is that the origin of a claim contains zero evidence as to that claim's truth.

I had a look back though your other posts and found this, which explains a lot, for me anyway. Most people would put some more options in there - yes, no, im pretty sure, its extremely unlikely etc..

Heres what I think is the problem, and why I think you need to change the way you are thinking - Your whole concept of what is "logical" or what is "using reason" seems to be constrained to what is formally known as deductive logic. You seem to have a really thorough understanding of this type of logic and have really latched on to it. Deductive logic is just a subset of logic. There is more to it than that.

I was searching for something to show you on other forms of logic and came across this book - "Probability Theory - The Logic of Science" Which looks awesome, Im going to read it myself, it gets great reviews. Ive only skimmed the first chapter... but that seems to be a good summary of how science works- why it does not use just deductive logic. Science draws most of its conclusions from probability, deductive logic is only appropriate in specific cases.

Conclusions based on probability - "Im pretty sure", "This is likely/unlikely" are extremely valid - and rational. Your forcing yourself to use deductive logic, and only deductive logic, where its inappropriate.

>You have no way of knowing, and finding out that this person regularly hallucinates them tells you nothing about their actual existence.

Yeah I think with the info you've said we have it would be to little to draw a conclusion or even start to draw one. Agreed. It wouldnt take much more info for us to start having a conversation about probabilities though - Say we had another person from the planet and he says its actually the red striped jagerwappas that are actually taking over - and that these two creatures are fundamentally incompatible. ie. if x exists y can't and vice-versa.

The pymc3 documentation is a good place to start if you enjoy reading through mini-tutorials: pymc3 docs

Also these books are pretty good, the first is a nice soft introduction to programming with pymc & bayesian methods, and the second is quite nice too, albeit targeted at R/STAN.

• Bayesian Methods for Hackers
  
• Doing Bayesian Data Analysis (Kruschke)

Mathematical Statistics and Data Analysis.

I learned from this textbook and have found it quite good. It's pretty expensive, but may be what you're looking for. I really don't know how much statistics your classes covered, but the table of contents should give you a good idea on what to expect.

I also had success with cheap supplemental books from Dover, which can cover quite a lot of undergraduate statistics at an affordable price. I found good use in Statistical Inference by Rohatgi.

I really love Probability Theory: The Logic of Science by Jaynes. While it is not a physics book, it was written by one. It is very well written, and is filled with common sense (which is a good thing). I really enjoy how probability theory is built up within it. It is also very interesting if you have read some of Jaynes' more famous works on applying maximum entropy to Statistical Mechanics.

Study what you find the most interesting!

Does your linear algebra include the spectral theorem or Jordan canonical form? IMHO, a pure math subject that is relatively the easiest to learn and is useful no matter what you do is linear algebra.

Group theory (representation theory) has also served me well so far.

If you want to learn GR and Hamiltonian mechanics in-depth, learning smooth manifolds would be a must. Smooth manifolds are basically spaces that locally look like Euclidean spaces and we can do calculus on. GR is on a pseudo-Riemannian manifold with changing metric (because of massive stuffs). Hamiltonian mechanics is on a cotangent bundle, which is a symplectic manifold (whereas Lagrangian mechanics is on a tangent bundle.) John Lee's book is a gentle starting point.

Edit: If you feel like the review of topology in the appendix is not enough, Lee also wrote a book on topological manifolds.

It is detailed. It just doesn't seem logical to me. His entire position is that since the odds against things being the way they are are so high, there must be a god that arranged them. It's a fundamental misunderstanding of probability. The chance of things being the way they are is actually 100%, because they are that way. We don't know how likely they were before they happened because we only have one planet and one solar system to examine. For all we know there could be life in most solar systems.

Even if that wasn't the case, even if we did have enough information to actually conclude that our existence here and now has a .00000000000000000000000000000001% chance of happening he then makes the even more absurd jump to saying "there being a God is the only thing that makes sense". God, especially the Christian God, is even less likely than the already unlikely chance of us existing at all. If it's extremely unlikely that we could evolve into what we are naturally, how is it less unlikely that an all-powerful, all-knowing, all-good being could exist for no discernible reason?

You should get him a copy of this book. It's great and will help him with these misconceptions. If you haven't read it, I highly suggest you do as well.

> Elementary Statistics
http://ftp.cats.com.jo/Stat/Statistics/Elementary%20Statistics%20%20-%20Picturing%20the%20World%20(gnv64).pdf

Presuming you mean this book, I am still at an absolute loss to understand how you think this doesn't somehow require algebra as a prerequisite.

All the manipulations about gaussian distributions, student t distributions, binomial distribution etc... or even the bit on regression, right there on page 502, how is that not algebra. It literally makes reference to the general form of a line in 2-space. Are they just expected to memorize those outright with no regard to their derivation?

How do you treat topics like expected value? Because it seems like right there on page 194 that they've given the general algebraic formula for discrete, real valued, random variable.

They seem to elide the treatment of continuous random variables. So I presume they won't even be going through the exercise of the mean of a Poisson.

All of that granted, this book still heavily relies on the ability to perform algebraic permutations. Right there on page 306 is the very z-score transform I explicitly mentioned earlier.

As far as where I teach, I don't, excepting the odd lecture to clients or coworkers. Typically, however, our domain does not fit prettily into the packaged up parameterized distributions of baccalaureate statistics. We deal in a lot of probabilistic graphical models, in manifold learning, in non-parametrics, etc.

The books I recommend to my audience (which is quite different than those who haven't a basic grasp on algebra) are:

• Hastie's book https://web.stanford.edu/~hastie/Papers/ESLII.pdf
  
• Boyd's book: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
  
• Papadimitriou's book: http://tocs.ulb.tu-darmstadt.de/116091606.pdf
  
• Gelman is a must read: https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954
  
• Rudin for the necessary background: https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf
  
• Liese: http://hbanaszak.mjr.uw.edu.pl/TempTxt/(Springer%20Series%20in%20Statistics)%20Klaus-J.%20Miescke,%20F.%20Liese%20(auth.)-Statistical%20Decision%20Theory_%20Estimation,%20Testing,%20and%20Selection%20-Springer-Verlag%20New%20York%20(2008).pdf
  
• Koller's book and course on PGMs
  
  and more...
  
  Now, this is not basic statistics. But I would also make a strong argument that the book you teach out of still requires knowledge of the treatment of basic symbolic manipulation--which is primarily taught in algebra.
  
  While I certainly have my own issues with the bottled up and black-box sort of way that frequentist statistics is taught to undergraduates, that is an entirely separate pedagogical and epistemological argument. I still just don't see how you can see the book out of which you teach as a replacement for algebra.

> There are some philosophical reasons and some practical reasons that being a "pure" Bayesian isn't really a thing as much as it used to be. But to get there, you first have to understand what a "pure" Bayesian is: you develop reasonable prior information based on your current state of knowledge about a parameter / research question. You codify that in terms of probability, and then you proceed with your analysis based on the data. When you look at the posterior distributions (or posterior predictive distribution), it should then correctly correspond to the rational "new" state of information about a problem because you've coded your prior information and the data, right?

Sounds good. I'm with you here.

> However, suppose you define a "prior" whereby a parameter must be greater than zero, but it turns out that your state of knowledge is wrong?

Isn't that prior then just an error like any other, like assuming that 2 + 2 = 5 and making calculations based on that?

> What if you cannot codify your state of knowledge as a prior?

Do you mean a state of knowledge that is impossible to encode as a prior, or one that we just don't know how to encode?

> What if your state of knowledge is correctly codified but makes up an "improper" prior distribution so that your posterior isn't defined?

Good question. Is it settled how one should construct the strictly correct priors? Do we know that the correct procedure ever leads to improper distributions? Personally, I'm not sure I know how to create priors for any problem other than the one the prior is spread evenly over a finite set of indistinguishable hypotheses.

The thing about trying different priors, to see if it makes much of a difference, seems like a legitimate approximation technique that needn't shake any philosophical underpinnings. As far as I can see, it's akin to plugging in different values of an unknown parameter in a formula, to see if one needs to figure out the unknown parameter, or if the formula produces approximately the same result anyway.

> read this book. I promise it will only try to brainwash you a LITTLE.

I read it and I loved it so much for its uncompromising attitude. Jaynes made me a militant radical. ;-)

I have an uncomfortable feeling that Gelman sometimes strays from the straight and narrow. Nevertheless, I looked forward to reading the page about Prior Choice Recommendations that he links to in one of the posts you mention. In it, though, I find the puzzling "Some principles we don't like: invariance, Jeffreys, entropy". Do you know why they write that?

I would try Mathematical Statistics and Data Analysis by Rice. The standard intro text for Mathematical Statistics (this is where you get the proofs) is Wackerly, Mendenhall, and Schaeffer but I find this book to be a bit too dry and theoretical (and I'm in math). Calculus is less important than a thorough understanding of how random variables work. Rice has a couple of pretty good chapters on this, but it will require some mathematical maturity to read this book. Good luck!

Depending on how strong your math/stats background is you might consider Statistical Inference by Casella and Berger. It's what we use for our first year PhD Mathematical Statistics course.

That might be a little too difficult if you're not very comfortable with probability theory and basic statistics. If you look at the first few chapters on Amazon and it seems like too much I recommend Mathematical Statistics and Data Analysis by Rice which I guess I would consider a "prequel" to the Casella text. I worked through this in an advanced statistics undergrad course (along with Mostly Harmless Econometrics and the Goldberger's course in Econometrics).

Let's see, if you're interested in Stochastic Models (Random Walks, Markov Chains, Poisson Processes etc), I recommend Introduction to Stochastic Modeling by Taylor and Karlin. Also something I worked through as an undergrad.

Bayes is the way to go: Ed Jayne's text Probability Theory is fundamental and a great read. Free chapter samples are here. Slightly off topic, David Mackay's free text is also wonderfully engaging.

I dove into this stuff almost two years ago with very little preparation or background. Now I'm in an MS program for Applied Statistics, and doing quite well. Here are some tips that worked for me:

• If you don't have time to back up and regroup, check out Khan Academy, and this guy's YouTube videos. These can help with specific concepts.
  
  
• If you have time to back up and regroup, check out Coursera, Udacity, EdX, and the other MOOCs. Coursera in particular has some very good courses dealing with statistics.
  
  
• Take a look at Statistics for Dummies and Naked Statistics.
  
  
• Use Reddit and StackOverflow. But use them wisely, and only after you've exhausted other means.
  
  Good luck.

The absolute best book I've found for someone with a frequentist background and undergraduate-level math skills is Doing Bayesian Data Analysis by John Kruschke. It's a fantastic book that goes into mathematical depth only when it needs to while also building your intuition.

The second edition is new and I'd recommend it over the first because of its improved code. It uses JAGS and STAN instead of Bugs, which is Windows-only now.

One of the post-docs in my department did his dissertation with Bayesian stats and he essentially had to teach himself! He strongly recommended this as a place to start if you are interested in that topic -- https://www.amazon.com/gp/product/1482253445/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1 (I have not read it yet.)

One of our computer science profs teaches Bayes for the CS folks and said he would be willing to put together a class for psych folks in conjunction with some other people, so that's a place where I am hoping to develop some competency at some point. I strongly recommend reaching outside of your department, especially if you are at a larger university!

R for Data Science is great, especially because it teaches tidyverse.

Another good book is Learning Statistics with R: A tutorial for psychology students and other beginners, which also teaches the implementation of basic statistical techniques, like ANOVA or linear regression.

If you have some time spare, you can follow it by Data Analysis Using Regression and Multilevel/Hierarchical Models, which is also (mostly) based on R.

The Visual Display of Quantitative Information is a good book on the principles of data visualization. It’s theoretical, so no R examples.

Complex Surveys: A Guide to Analysis Using R is great if you work with survey data, especially if you work with complex designs (which nowdays is pretty much all the time).

Personaly, I would also invest some time learning methodology. Sadly, I can’t help you here, because I didn’t used textbook for this, but people seem to like books from Earl Babbie.

I've wasted too much time trying to find the so-called "right" statistics book. I'm still early in my journey, going through calculus using Prof. Leonards videos while working through a Linear Algebra book all in prep for tackling a stats book. Here's a list of books that I've had a look at so far.

​

• Probability and Statistical Inference by Hogg, Tanis and Zimmerman
  
• Mathematical Statistics with Applications by Wackerly
  
  These seem to be of a similar level with regards to rigour, as they aren't that rigourous. That's not to say you can get by without the calculus prereq and even linear algebra
  
  ​
  
  The other two I've been looking at which seem to be a lot more complex are
  
  
• Introduction to Mathematical Statistics by Hogg as well. I'd think it's the more rigorous version of the book mentioned above by the same author
  
• All of Statistics by Wasserman which seems to require a lot of prior knowledge in statistics, but I think tackles just the perfect topics for machine learning
  
  And then there's Casella and Berger's Statistical inference, which I looked at once and decided not to look at again until I can manage at least one of the aforementioned books. I think I'm leaning most to the first book listed. Whichever one you decide to use, good luck with your journey.
  
  ​

For probability I'd recommend Introduction to Probability Theory by Hoel, Port & Stone. It has the best explanations of any probability book I've seen, great examples, and answers to most of the problems are in the back (making it well-suited for self-study). I think it's still the best introductory book on the subject, despite its age. Amazon has used copies for cheap.

For statistics, you have to be more precise as to what you mean by an "average undergraduate statistics" course. There's a difference between the typical "elementary statistics" course and the typical "mathematical statistics" course. The former requires no calculus, but goes into more detail about various statistical procedures and tests for practical uses, while the latter requires calculus and deals more with theory than practice. Learning both wouldn't be a bad idea. For elementary stats there are lots of badly written books, but there is one jewel: Statistics by Freedman, Pisani & Purves. For mathematical statistics, Introduction to Mathematical Statistics by Hogg & Craig is decent, though a bit dry. I don't think that Statistical Inference by Casella & Berger is really any better. Those are the two most-used textbooks on the subject.

You received A's in your math classes at a major public university, so I think you're in pretty good shape. That being said, have you done proof-based math? That may help tremendously in giving intuition because with proofs, you are giving rigor to all the logic/theorems/ formulas, etc that you've seen in your previous math classes.

Statistics will become very important in machine learning. So, a proof-based statistics book, that has been frequently recommended by /r/math and /r/statistics is Statistical Inference by Casella & Berger: https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126

I've never read it myself, but skimming through some of the beginning chapters, it seems pretty solid. That being said, you should have an intro to proof-course if you haven't had that. A good book for starting proofs is How to Prove It: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995

Discrete mathematics and any proof based math in general is what college based math should be like- if you continue to take upper level math and CS courses, you will undoubtedly face this style of math again. Plug and chug (which is what a lot of calculus is) will no longer be the norm.

There is often a very large learning curve for students who are not used to seeing this type of math- so don't stress out too much about it. Eventually, you'll break a point where everything will make (sort of) sense. I went through the exact same thing when I took discrete for the first time, and I felt like I was getting destroyed on everything (I still suck at some topics) until I suddenly hit a point of clarity where I could see how most topics were tied in together. Mathematics, and especially an introductory discrete course, is cruel in that way- that every topic you learn is inherently related to each other, so if you already fall behind just a little, the mountain to catch up just becomes incredibly massive incredibly fast- and it's hard to even pinpoint a place to even start to catch up.

You may be lost in learning elementary proof techniques, or number theory, and then the next topic (say it's graph theory) utilizes a bunch concepts and previous proofs from number theory, and then the next topic might use something proved in graph theory and number theory, and so on. All of a sudden, nothing makes sense, and to learn topic ___, you need to know graph theory, but to know graph theory, you need to know number theory, but you don't know number theory that well, and some topics in number theory can perhaps be explained by another topic in graph theory (or any topic for that matter) The chain is all interlinked and it may difficult to even see where to start- but it is for this reason that once you cross this steep barrier, most things will suddenly become clear to you.

So I'd advise you to just continue visiting professor office hours, asking more questions, asking for other students' help, doing more and more practice. It may seem like you're getting nowhere, but you're essentially learning a new language right now, so it'll obviously take sometime until you feel as if you know what you're doing. Figuring out where people get the intuition to suggest seemingly random functions or a set of numbers or some assumption will come to you slowly, and slowly you'll break more and more of this chain.

https://www.amazon.com/Discrete-Mathematics-Laszlo-Lovasz/dp/0387955852 is another book my professor enjoyed using as a supplmenet.

At the moment, psychology is largely ad-hoc, and not a modicum of progress would've been made without the extensive utilization of statistical methods. To consider the human condition does not require us to simply extrapolate from our severely limited experiences, if not from the biases of limited datasets, datasets for which we can't even be certain of their various e.g. parameters etc..

For example, depending on the culture, the set of phenotypical traits with which increases the sexual attraction of an organism may be different - to state this is meaningless and ad-hoc, and we can only attempt to consider the validity of what was stated with statistical methods. Still, there comes along social scientists who would proclaim arbitrary sets of phenotypical features as being universal for all humans in all conditions simply because they were convinced by limited and biased datasets (e.g. making extreme generalizations based on the United States' demographic while ignoring the entire world etc.).

In fact, the author(s) of "Probability Theory: The Logic of Science" will let you know what they think of the shaky sciences of the 20th and 21st century, social science and econometrics included, the shaky sciences for which their only justifications are statistical methods.
_

With increasing mathematical depth and the increasing quality of applied mathematicians into such fields of science, we will begin to gradually see a significant improvement in the validity of said respective fields. Otherwise, currently, psychology, for example, holds no depth, but the field itself is very entertaining to me; doesn't stop me from enjoying Michael's "Mind Field" series.

For mathematicians, physics itself lacks rigour, let alone psychology.
_

Note, the founder of "psychoanalysis", Sigmund Freud, is essentially a pseudo-scientist. Like many social scientists, he made the major error of extreme extrapolation based on his very limited and personal life experiences, and that of extremely limited, biased datasets. Sigmund Freud "proclaimed" a lot of truths about the human condition, for example, Sigmund Fraud is the genius responsible for the notion of "Penis Envy".

In the same century, Einstein would change the face of physics forever after having published the four papers in his miracle year before producing the masterpiece of General Relativity. And, in that same century, incredible progress such that of Gödel's Incompleteness Theorem, Quantum Electrodynamics, the discovery of various biological reaction pathways (e.g. citric acid cycle etc.), and so on and so on would be produced while Sigmund Fraud can be proud of his Penis Envy hypothesis.

I've used a book by Gelman for self study. Great author, very good at using meaningful graphics -- which may be an effective way to convey ideas to students.

If you liked Consider the Lobster, then you will also very probably like A Supposedly Fun Thing I'll Never Do Again and Both Flesh and Not.

Edited to add that Everything and More is also very good, though it's not a collection of essays.

By introductory, do you mean undergrad level and advanced do you grad level?

If that is the case: The most widely used undergrad book is Wackerly et al. I also taught out of Devore before and it is not bad.

Wackerly covers more topics, but does so in a much more terse manner. Devore covers things better, but covers less things (some of which are pretty important).

Grad: Casella and Berger. People might have their qualms with this book but there is really no better book out there.

Both Lee's and Tu's books are on my reading list. They both seem excellent.

However, my vote is for Professor Tu's book, mainly because it manages to get to some of the big results more quickly, and he evidently does so without a loss of clarity. In the preface to the first edition, he writes "I discuss only the irreducible minimum of manifold theory that I think every mathematician should know. I hope that the modesty of the scope allows the central ideas to emerge more clearly." Consequently, his book is roughly half the length of Lee's.

I'd rather hit the most essential points first, and then if I want a more expansive view, I'd pick up Lee.

Disclaimer: I may not participate very frequently, as I have some other irons in the fire, so you might want to weigh my vote accordingly. If your sub sticks around for a while, I'd definitely like to join in when I can.

It sounds like you would enjoy Everything and More: A Compact History of Infinity, also by DFW. Fascinating read, non-fiction, both somewhat technical and easily readable.

"The task Wallace has set himself is enormously challenging: without radically compromising the complexity of the philosophy, metaphysics, or mathematics that underlies the evolving concept of infinity, present the material to a lay audience in a manner that is entertaining."

https://www.amazon.com/Everything-More-Compact-History-Infinity/dp/0393339289

"Bayesian" is a very very vague term, and this article isn't talking about Bayesian networks (I prefer the more general term graphical models), or Bayesian spam filtering, but rather a mode of "logic" that people use in everyday thinking. Thus the better comparison would be not to neural nets, but to propositional logic, which I think we can agree doesn't happen very often in people unless they've had lots of training. My favorite text on Bayesian reasoning is the Jaynes book..

Still, I'm less than convinced by the representation of the data in this article. Secondly, the article isn't even published yet to allow anyone to review it. Thirdly, I'm suspicious of any researcher that talks to the press before their data is published. So in short, the Economist really shouldn't have published this, and should have waited. Yet another example of atrocious science reporting.

I have very few universal recommendations. Think the only one that actually comes to mind is "Introduction to Probability" by Blitzstein and Hwang. It is probably the best book on probability that I've found for a broad audience. It also has a corresponding video lecture series.

If you want any more, please answer this:

• What is your interest?
  
• What is your background?
  
• What do you want to learn to do?
  
  Maybe I can see what I have laying around that meets your criteria.
  

Just completed Probability this semester, and moving on to Statistical Inference next semester. Calc. B is a prerequisite, and wound up seeing plenty of it along with a little Calc C (just double integrals). I'm an Applied Mathematics undergrad major btw and former Physics major from some years ago. I wound up enjoying it despite my bad attitude in the beginning. I keep hearing from fellow math majors that Statistical Inference is really difficult. Funny thing is I heard the same about Linear Algebra and didn't find it overwhelming. I'll shall soon find out. We used Wackerly's Mathematical Statistics with Applications. I liked the book more than most in my class. Some thought it was overly complicated and didn't explain the content well. Seems I'm always hearing some kind of complaint about textbooks every semester. Good luck.

Somewhat facetiously, I'd say the probability that an individual who has voted in X/12 of the last elections will vote in the next election is (X+1)/14. That would be my guess if I had no other information.

As the proverb goes: it's difficult to make predictions, especially about the future. We don't have any votes from the next election to try to discern what relationship those votes have to any of the data at hand. Of course that isn't going to stop people who need to make decisions. I'm not well-versed in predictive modeling (being more acquainted with the "make inference about the population from the sample" sort of statistics) but I wonder what would happen if you did logistic regression with the most recent election results as the response and all the other information you have as predictors. See how well you can predict the recent past using the further past, and suppose those patterns will carry forward into the future. Perhaps someone else can propose a more sophisticated solution.

I'm not sure how this data was collected, but keep in mind that a list of people who have voted before is not a complete list of people who might vote now, since there are some first-time voters in every election.

If you want to get serious about data modeling in social science, you might check out this book by statistician/political scientist Andrew Gelman.

> I'm hoping for something like what Div, Grad, Curl and All That does for Vector Calculus.

Is that a math text? I am not really familiar with it, but from what I heard it sounds more like a physics/engineering text. Does it have any formal proofs in it?

You won't be able to get too far with a proofless(?) Abstract Algebra text if there exists one to begin with. Even Charles Pinter's A Book of Abstract Algebra presupposes some degree of mathematical maturity.

Anyway, try these and see if you like them:


Visual Group Theory by Nathan Carter


Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem by Al Cuoco, Joseph J. Rotman

The short version is that in a bayesian model your likelihood is how you're choosing to model the data, aka P(x|\theta) encodes how you think your data was generated. If you think your data comes from a binomial, e.g. you have something representing a series of success/failure trials like coin flips, you'd model your data with a binomial likelihood. There's no right or wrong way to choose the likelihood, it's entirely based on how you, the statistician, thinks the data should be modeled. The prior, P(\theta), is just a way to specify what you think \theta might be beforehand, e.g. if you have no clue in the binomial example what your rate of success might be you put a uniform prior over the unit interval. Then, assuming you understand bayes theorem, we find that we can estimate the parameter \theta given the data by calculating P(\theta|x)=P(x|\theta)P(\theta)/P(x) . That is the entire bayesian model in a nutshell. The problem, and where mcmc comes in, is that given real data, the way to calculate P(x) is usually intractable, as it amounts to integrating or summing over P(x|\theta)P(\theta), which isn't easy when you have multiple data points (since P(x|\theta) becomes \prod_{i} P(x_i|\theta) ). You use mcmc (and other approximate inference methods) to get around calculating P(x) exactly. I'm not sure where you've learned bayesian stats from before, but I've heard good things , for gaining intuition (which it seems is what you need), about Statistical Rethinking (https://www.amazon.com/Statistical-Rethinking-Bayesian-Examples-Chapman/dp/1482253445), the authors website includes more resources including his lectures. Doing Bayesian data analysis (https://www.amazon.com/Doing-Bayesian-Data-Analysis-Second/dp/0124058884/ref=pd_lpo_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=58357AYY9N1EZRG0WAMY) also seems to be another beginner friendly book.

By the way, do you know if things like linear/nonlinear regression, ANOVA and multivariate statistics is useful for me? Like stuff from https://www.amazon.com/Applied-Linear-Statistical-Models-Michael/dp/007310874X/ref=dp_ob_title_bk or https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/ref=pd_sim_14_8?_encoding=UTF8&pd_rd_i=1439840954&pd_rd_r=c9c0f3c5-f332-11e8-9f0a-0f336f5f387a&pd_rd_w=xHkKH&pd_rd_wg=7lXm5&pf_rd_i=desktop-dp-sims&pf_rd_m=ATVPDKIKX0DER&pf_rd_p=18bb0b78-4200-49b9-ac91-f141d61a1780&pf_rd_r=PFCZ1JM04FMAVAHG6VNP&pf_rd_s=desktop-dp-sims&pf_rd_t=40701&psc=1&refRID=PFCZ1JM04FMAVAHG6VNP

​

You can tell how little I know since I'm kinda shooting topics at the wall hoping something sticks

It is hard to provide a "comprehensive" view, because there's so much disperate material in so many different fields that draw upon probability theory.

Feller is an approachable classic that covers all of the main results in traditional probability theory. It certainly feels a little dated, but it is full of the deep central limit insights that are rarely explained in full in other texts. Feller is rigorous, but keeps applications at the center of the discussion, and doesn't dwell too much on the measure-theoretical / axiomatic side of things. If you are more interested in the modern mathematical theory of probability, try Probability with Martingales.

On the other hand, if you don't care at all about abstract mathematical insights, and just want to be able to use probabilty theory directly for every-day applications, then I would skip both of the above, and look into Bayesian probabilistic modelling. Try Gelman, et. al..

Of course, there's also machine learning. It draws on a lot of probability theory, but often teaches it in a very different way to a traditional probability class. For a start, there is much more emphasis on multivariate models, so linear algebra is much more central. (Bishop is a good text).

I learned from Wackerly which is decent, though I think Devore's presentation is better, but not as deep. Both have plenty of exercises to work with.

Casella and Berger is the modern classic, which is pretty much standard in most graduate stats programs, and I've heard good things about Stat Labs, which uses hands-on projects to illuminate the topics.

The single best resource for 103/104 is The Calculus Lifesaver by Adrian Banner. There's a book and a series of recorded review sessions. I stopped showing up to 104 lectures when I found these because they were so much more thorough than the classes. Banner also did review sessions for 201/202 when you reach that point, which are equally good.

This book and his videos: https://www.amazon.com/Calculus-Lifesaver-Tools-Princeton-Guides/dp/0691130884

I was good at calculus, but this book made anything I struggled to fully understand much easier. He does a good job of looking back at how previous work supports and and talks about how this relates to future topics.

Yes, -5/9 is a typo, just as you say.

By the way, the lecturer in the MIT video is Gilbert Strang, and his textbook, Introduction to Linear Algebra is the text that he uses for the course. I'm not really familiar with that book, but I believe that it has a pretty good reputation. See for example, this recent reddit thread, where Strang is mentioned several times.

To piggy back off of danielsmw's answer...

> Fourier analysis is used in pretty much every single branch of physics ever, seriously.

I would phrase this as, "partial differential equations (PDE) are used in pretty much every single branch of physics," and Fourier analysis helps solve and analyze PDEs. For instance, it explains how the heat equation works by damping higher frequencies more quickly than the lower frequencies in the temperature profile. In fact Fourier invented his techniques for exactly this reason. It also explains the uncertainty principle in quantum mechanics. I would say that the subject is most developed in this area (but maybe that's because I know most about this area). Any basic PDE book will describe how to use Fourier analysis to solve linear constant coefficient problems on the real line or an interval. In fact many calculus textbooks have a chapter on this topic. Or you could Google "fourier analysis PDE". An undergraduate level PDE course may use Strauss' textbook whereas for an introductory graduate course I used Folland's book which covers Sobolev spaces.

If you wanted to study Fourier analysis without applying it to PDEs, I would suggest Stein and Shakarchi or Grafakos' two volume set. Stein's book is approachable, though you may want to read his real analysis text simultaneously. The second book is more heavy-duty. Stein shows a lot of the connections to complex analysis, i.e. the Paley-Wiener theorems.

A field not covered by danielsmw is that of electrical engineering/signal processing. Whereas in PDEs we're attempting to solve an equation using Fourier analysis, here the focus is on modifying a signal. Think about the equalizer on a stereo. How does your computer take the stream of numbers representing the sound and remove or dampen high frequencies? Digital signal processing tells us how to decompose the sound using Fourier analysis, modify the frequencies and re-synthesize the result. These techniques can be applied to images or, with a change of perspective, can be used in data analysis. We're on a computer so we want to do things quickly which leads to the Fast Fourier Transform. You can understand this topic without knowing any calculus/analysis but simply through linear algebra. You can find an approachable treatment in Strang's textbook.

If you know some abstract algebra, topology and analysis, you can study Pontryagin duality as danielsmw notes. Sometimes this field is called abstract harmonic analysis, where the word abstract means we're no longer discussing the real line or an interval but any locally compact abelian group. An introductory reference here would be Katznelson. If you drop the word abelian, this leads to representation theory. To understand this, you really need to learn your abstract/linear algebra.

Random links which may spark your interest:

• Fourier analysis also finds applications in number theory, for example in the Hardy-Littlewood circle method.
  
• Solving cubic equation using discrete Fourier transform
  
• The Wavelet transform
  
• Fourier optics
  
• Monstrous moonshine
  
• Fractional Laplacian/calculus

IQ tests are calibrated to return an average of 100. Absent any evidence to the contrary, we assume the null hypothesis and place all subgroups at the global average. Here's a good starting point if you're interested in learning more.

Educational barriers for African Americans are well documented and muddy the relationship between intelligence and education. You'll note that the educational data I provided earlier was solely for whites, where the relationship is clear cut. I'm unaware of any reliable data for blacks.

Now, since you're presumably white, and we do have that data, would you mind telling the audience a little about your education, and we'll see what inferences we can draw?

it most certainly is! There's a whole approach to statistics based around this idea of updating priors. If you're feeling ambitious, the book Probability theory by Jaynes is pretty accessible.

It depends on what kind of math you want to learn. If you want to get up to speed on your basic math, khan academy is the way to go. However, I think that is probably a waste of your time. The math that you will see in high school and the first year or too of college has very little to do with what a mathematician might consider 'real math.' Frankly I found it boring as hell and I majored in math undergrad and grad.

If I were you, I would start with something interesting and if you end up really liking math, go back and pick up algebra and calculus. So check out the two books below:

This book will walk you through really high level stuff in an easy to understand way. As a grad student I would hang out in this class because it was rather fun.

This book is a history of math/pop math book. As an undergrad it put the field into perspective. Lots and lots of really useful information for anyone, especially someone who is interested in being well learned.

Interdisciplinary connections spring up from generality. You'd be hard pressed to find a spontaneous connection between something like particle phenomenology and an unrelated field.

To illustrate this idea of generality, consider the methods of statistical mechanics, which are so general that they can be used to describe everything from black holes to ferromagnets. However, the methods have also been used to model neural networks and social dynamics (the latter being accurate enough to successfully recreate historical events.)

What makes statistical mechanics more general than other branches? Probably the fact that it's almost more mathematics than physics, specifically a branch of probability theory regarding highly correlated random variables.

With this in mind, perhaps you'd benefit from focusing your attention on the mathematical ideas that drive physics rather than physics itself. Take the calculus of variations which, whilst developed for problems in classical mechanics, has found applications in mathematical optimisation. Another example being brownian motion, the mathematics of which have been generalised to higher dimensions and applied to finance. The mathematics behind relativity is differential geometry, which has been applied to too many fields to list.

I'd recommend having a look at Mathematical Methods for Physicists by Arfken, Weber and Harris for a broad overview of the methods.

In the lead up to calc first thing you want to do is just make sure you're algebra skills are pretty solid. A lot of people neglect it and then find the course to be harder than it needed to be because you really use algebra throughout.

Beyond that, if you want an extra book to study with and get practice problems from The Calculus Lifesaver is a big book of calculus you can use from now and into a first year college calculus course. If you do get it, don't worry about reading the whole thing from cover to cover, or doing all of the problems in it. It is a big book for a reason, it definitely covers more than you need to know for now, so don't get overwhelmed, it all comes with time.

Best of luck

I'm really fond of Jaynes' Probability Theory: The Logic of Science and Rudin's Principles of Mathematical Analysis. Both are excellent, clearly written books in their own way.

I'd recommend Discrete Mathematics, Elementary and Beyond By Lovász, Pelikán, and Vesztergombi. It's the book I'm using in my undergraduate discrete math course, and I think it's a great introductory book that explores many areas of discrete math, and should allow you to see which field interests you most.

Casella and Berger is one of the go-to references. It is at the advanced undergraduate/first year graduate student level. It's more classical statistics than data science, though.

Good statistical texts for data science are Introduction to Statistical Learning and the more advanced Elements of Statistical Learning. Both of these have free pdfs available.

As long as you have a solid foundation in algebra (and basic trig), you should be fine. However, you have to put in the study time. If you want supplementary material, I'd recommend The Calculus Lifesaver, which was a tremendous help for me, although it only covers single-variable calculus (i.e., Calc I and II). The cool thing about this book is that its author (a Princeton University professor) also has video lectures posted online.

This is the basis of calculus. An infinitesimal (1/Infinity) can both = 0 and > 0. When calculus was first presented to the math community, they saw this and called it a bunch of liberal hippie bullshit. It took ~100-150 years for calculus to be fully formalized and accepted within the math community, but it was immediately accepted in the engineering community because it worked. If you’re interested, I highly recommend Everything and More: A Compact History of Infinity by David Foster Wallace

Casella & Berger is the go-to reference (as Smartless has already pointed out), but you may also enjoy Jaynes. I'm not sure I'd say it's quick but if gaps are your concern, it's pretty drum-tight.

One way is picking a distribution with a mode and a "concentration" around that mode that reflects what you have. John Kruschke does an amazing job at explaining how to pick Beta priors based off of that in Doing Bayesian Data Analysis (which, may I note, has the best cover of any statistics book I've ever read).

• How to Lie with Statistics by Darrell Huff is a quick and fun read that will sharpen your ability to identify fraudulent claims.
  
  
• My favorite stats book for people who are not math-geeks is Statistical Methods for Psychology by David Howell
  
  
• The particular technique used in the paper is somewhat beyond beginner level. A third good book to read might be Data Analysis Using Regression and Multilevel/Hierarchical Models by Andrew Gelman

this book really helped me in undergrad. Has a lot of really good concepts. It went along with a course but it does a great job on its own explaining some of the most relevant concepts to computer science.

So I am a political scientist (though my research crosses into sociology).

What I would recommend is starting by learning Generalized Linear Models (GLMs). Logistic regression is one type, but GLMs are just a way of approaching a bunch of other type of dependent variables.

Gelman and Hill's book is probably the best single text book that can cover it all. I think it provides examples in R so you could also work on picking up R. It covers GLMs and multi-level models which are also relatively common in sociology.

If you really need it dumbed down, I would recommend Asimow and Maxwell. This text has a solutions manual. Note that this is specifically tailored toward actuarial exams - i.e., people that have to learn the material quickly but not necessarily for grad school. (And yes, the website is legit. I've done some contract work for them in the past and have ordered books through them.)

If you don't mind something more mathematical, I would recommend Wackerly et al.

Hey I'm a physics BSc turned mathematician.

I would suggest starting with topology and functional analysis. Functional analysis is the foundation of quantum mechanics, and topology is necessary to properly understand manifolds, which are the foundation of relativity.

I would suggest Kreyszig for functional analysis. It's probably the most gentle functional analysis book out there.

For topology, I would suggest John Lee. This topology text is unique because it teaches general topology with a view towards manifolds. This makes it ideal for a physicist. If you want to know about Lie algebras and Lie groups, the sequel to this text discusses them.

Of course efforts like this won't fly because there will be people who sincerely want to can them because it's "computerized racial profiling," completely missing the point that, if race does correlate with criminal behavior, you will see that conclusion from an unbiased system. What an unbiased system will also do is not overweight the extent to which race is a factor in the analysis.

Of course, the legitimate concern some have is about the construction of prior probabilities for these kinds of systems, and there seems to be a great deal of skepticism about the possibility of unbiased priors. But over the last decade or two, the means of constructing unbiased priors have become rather well understood, and form the central subject matter of Part II of E.T. Jaynes' Probability Theory: The Logic of Science, which I highly recommend.

Gelman's book is pretty awesome. http://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Models/dp/052168689X/ref=pd_sim_b_1

A lot of the recommendations in this thread are good, I'd like to add "Bayesian Data Analysis 3rd edition" by Gelman et al. Useful if you encounter Bayesian models, especially hierarchical/multilevel models.

Your professors really aren't expecting you to reinvent groundbreaking proofs from scratch, given some basic axioms. It's much more likely that you're missing "hints" - exercises often build off previous proofs done in class, for example.

I appreciated Laura Alcock's writings on this, in helping me overcome my fear of studying math in general:
https://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/

https://www.amazon.com/dp/0198723539/ <-- even though you aren't in analysis, the way she writes about approaching math classes in general is helpful

If you really do struggle with the mechanics of proof, you should take some time to harden that skill on its own. I found this to be filled with helpful and gentle exercises, with answers: https://www.amazon.com/dp/0989472108/ref=rdr_ext_sb_ti_sims_2

And one more idea is that it can't hurt for you to supplement what you're learning in class with a more intuitive, chatty text. This book is filled with colorful examples that may help your leap into more abstract territory: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X

> the first half of my degree was heavy on theoretical statistics,

Really? Wow, I'm impressed. Actual coverage of even basic theoretical stats is extremely rare in psych programs. Usually it's a bunch of pronouncements from on high, stated without proof, along with lists of commandments to follow (many of dubious value) and a collection of bogus rules of thumb.

What book(s) did you use? Wasserman? Casella and Berger? Cox and Hinkley? or (since you say it was heavy on theory) something more theoretical than standard theory texts?

I'd note that reaction times (conditionally on the IVs) are unlikely to be close to normal (they'll be right skew), and likely heteroskedastic. I'd be inclined toward generalized linear models (perhaps a gamma model -probably with log-lnk if you have any continuous covariates- would suit reaction times?). And as COOLSerdash mentions, you may want a random effect on subject, which would then imply GLMMs

For mathematical statistics: Statistical Inference.

Bioinformatics and Statistics: Statistical Methods in Bioinformatics.

R: R in a Nutshell.

Edit: The Elements of Statistical Learning (free PDF!!)

ESL is a great book, but it can get very difficult very quickly. You'll need a solid background in linear algebra to understand it. I find it delightfully more statistical than most machine learning books. And the effort in terms of examples and graphics is unparalleled.

> I'd like to know, how did you learn to use R?

My batshit crazy lovable thesis advisor was teaching intro datascience in R.

He can't really lecture and he have high expectation. The class was for everybody including people that don't know how to program. The class book was advance R http://adv-r.had.co.nz/... (red flag).

We only survived this class because I had a cs undergrad background and I gave the class a crash course once. Our whole class was more about how to implement his version of random forest.

I learned R because we had to implement a version of Random forest with Rpart package and then create a package for it.

Before this a dabble in R for summer research. It was mostly cleaning data.

So my advice would be to have a project and use R.

>how did you learn statistics?

Master program using the wackerly book and chegg/slader. (https://www.amazon.com/Mathematical-Statistics-Applications-Dennis-Wackerly/dp/0495110817)

It's a real grind. You need to learn probability first before even going into stat. Wackerly was the only real book that break down the 3 possible transformations (pdf,cdf, mgf).

I can recommend a very good book, I am using it and it is beautiful.

Only if you are too dumb to know how to use it. Knowing a median is actually quite useful when making future predictions.

I would encourage you to read up on statistics, so you can focus on things that matter, rather than on the odds that Urban Meyer wins a game when there is snow within 100 miles and he is wearing khakis.

http://www.amazon.com/Statistics-Dummies-Deborah-J-Rumsey/dp/0470911085

I’d recommend Blitzstein’s Into to Probability book- it’s the book used for Harvard’s Stat110 which has free lectures online as well.

https://www.amazon.com/Introduction-Probability-Chapman-Statistical-Science/dp/1466575573

I looked at similar (WA resident also) but there's only a few community college classes that are interesting (linear algebra, probability, ODE) so then you're looking at UW/WSU tuition. There's a couple applied tracks you could consider: machine learning and financial math:

https://metacademy.org/roadmaps/

http://www.deeplearningweekly.com/pages/open_source_deep_learning_curriculum

https://www.quantstart.com/articles/Quantitative-Finance-Reading-List

-----------

Self study: math for physics texts like Arfken/Harris/Weber, Boas, Riley/Hobson, Thomas Garrity

http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmaster.pdf

https://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544

How would you rather split beginner vs intermediate/advanced ?

My feeling was that Ben Lambert's book would be a good intro and that Bayesian Data Analysis would be a good next ?

So off the top off my head, I can’t think of any courses. Here are three books that include exercieses that are relatively comprehensive and explain their material well. They all touch upon basic methods that are good to know but also how to do analyses with them.

• Hands-On Machine Learning with Scikit-Learn and TensorFlow. General machine learning and intro to deep learning (python) - link
  
  
• The Elements of Statistical Learning. Basic statistical modelling (R) link
  
  
• Statistical rethinking. Bayesian statistics (R) link - there are lectures to this book as well
  
  But there are many many others.
  
  Then there are plenty of tutorials to python, R or how to handle databases (probably the core programming languages, unless you want to go the GUI route).
  
  

https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X

I'm not much of a visual person so I didn't learn much from skimming it, but many people stand by it and it's a beautiful exposition (of a topic I also think is beautiful) indeed!

In all seriousness, the applications of analysis to geometry can be really interesting and insightful, but to get to them you would have to first have background in differential topology, which it seems you lack. That might be a good subject to start with. A good book would be John Lee's An Introduction to Smooth Manifolds.

Lee is still the easiest and best for self study. https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395/ref=sr_1_3?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-3

followed by:

https://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/1441999817/ref=sr_1_2?keywords=lee+manifolds&qid=1558265795&s=gateway&sr=8-2

​

It's long, taking almost two volumes to get to Stokes theorem. If conciseness is important, you can just read Warner:

https://www.amazon.com/Foundations-Differentiable-Manifolds-Graduate-Mathematics/dp/0387908943/ref=sr_1_1?keywords=Warner+manifolds&qid=1558265966&s=gateway&sr=8-1

​

Tu looks good, but I haven't read it carefully.

Rather than list various courses, I'll say this. If you can use all the techniques in this book:

http://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Edition/dp/0123846544/ref=dp_ob_title_bk/185-3957242-1103639

and understand the content of this book:

http://www.amazon.com/Mathematical-Physics-Sadri-Hassani/dp/0387985794/ref=sr_1_1?s=books&ie=UTF8&qid=1335192374&sr=1-1

then you will almost certainly know all they math you'll ever need for advanced undergraduate and general graduate courses. In fact, you'll almost certainly know much more than you'll need.

That's not to say that you should simply study those books - the second one is a gem, but the first is.... polarizing - but they're useful guides of what you ought to know.

Innumeracy by Paulos

A great read that deal in part with the general acceptability math incompetence has compared to other subjects. Also a fun book as a "math person" just in the way he speaks and confides in the reader.

http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Consequences-Vintage/dp/0679726012

Innumeracy is pretty entertaining (and useful), even if you're not a math person. It's only about 150 pages, so it's a quick read.

I personally think you should brush up on frequentist statistics as well as linear models before heading to Bayesian Statistics. A list of recommendations directed at your background:

• For Frequentist Statistics and Probability: Introduction to Probability, Mathematical Statistics With Applications or since you are more interested in applications of Stats to Machine Learning, All of Statistics.
  
  
• For Bayesian Statistics, I would start with Statistical Rethinking and Introduction to Bayesian Statistics for an intro.
  
  

Since you are already going to take Machine Learning and want to build a good statistical foundation, I highly recommend Mathematical Statistics with Applications by Wackerly et al.

I've heard good things about this book.

Edit: Here's the Canadian site: http://www.amazon.ca/Probability-Theory-The-Logic-Science/dp/0521592712

For some reason the US site doesn't have the table of contents.

Another handy link: http://bayes.wustl.edu/etj/prob/book.pdf

It contains the first 3 chapters, you can read through it and see if the explanations are to your satisfaction.

FYI, Jaynes actually wrote a whole probability textbook that essentially put together all his thoughts about probability theory. I haven't read it, but many people say it got some good stuff.

Only tangentially relevant, but a really good read!

Innumeracy

The Nature of Computation

(I don't care for people who say this is computer science, not real math. It's math. And it's the greatest textbook ever written at that.)

Concrete Mathematics

Understanding Analysis

An Introduction to Statistical Learning

Numerical Linear Algebra

Introduction to Probability

I'd suggest MATP 4600, Probability Theory & Applications. Only prerequisite is Calc if I remember right.

Or if you're confident in your time management, maybe read this textbook on your own; it's pretty accessible: https://www.amazon.com/gp/aw/d/0521592712/

(Neither of these will teach you a bunch of statistical tests, but those are easy to abuse if you don't understand the fundamentals ... and very easy to look up if you do understand the fundamentals.)

Analyzing Baseball with R is the best book, I believe:

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229

I also would download PitchRX and Baseball on a Stick to round out your toolkit!

-Kyle

Lewin's book is... interesting. It's a rare kind of music theory book which involves some actual math (with absolutely abhorrent typesetting). But it's actually quite straightforward if you know some basic group theory.

So I recommend to take a look at Visual Group Theory lectures on youtube: https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv. By the way, there's also a book with the same name.

Depends what your goal is. As you have a good background, I would not suggest any stats book or deep learning. First, read trough Probability theory - The logic of science and the go for Bishop's Pattern Recognition or Barbers's Bayesian Reasoning and ML. If you understand the first and one of the second books, I think you are ready for anything.

Since you're an applied math PhD, maybe the following are good. They are not applied though.

This is the book for first year statistics grad students at OSU.
http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126/ref=sr_1_1?ie=UTF8&qid=1368662972&sr=8-1&keywords=casella+berger

But, I like Hogg/Craig much more.
http://www.amazon.com/Introduction-Mathematical-Statistics-7th-Edition/dp/0321795431/ref=pd_sim_b_2

I believe each can be found in international editions, and for download on the interwebs.

Argh. Numerology. It's like every logical fallacy for numbers rolled into one.

I highly recommend you pick up a copy of Innumeracy.

Mostly because I wanted to analyze baseball stats, and at the time (4-5 years ago) that was mostly done in R. If the last industry conference I went to is any indication, it still is, many of the presentations features plots that were clearly ggplot2. There are also books like this one floating around: https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229/ref=nodl_.

I have never argued that draw is good for the game. If you read my posts around this subreddit, I have critized mojang for not putting in the proper way to the hand limit and have argued that it makes the game less tactical on several occasions. As for the rest, try this:
http://www.amazon.co.uk/Statistics-For-Dummies-Deborah-Rumsey/dp/0470911085

If you like logic and the scientific method, I recommend E. T. Jaynes' Probability Theory: The Logic of Science. You can buy it here:
http://www.amazon.com/Probability-Theory-The-Logic-Science/dp/0521592712/

or read a PDF here:
http://shawnslayton.com/open/Probability%2520book/book.pdf

There seem to be a few options. I've had this and this on my reading list for a while, but haven't got further than that.

I'm also interested in recommendations.

I've always loved Andrew Gelman and Jennifer Hill's book: http://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Models/dp/052168689X/ref=sr_1_1?s=books&ie=UTF8&qid=1410039864&sr=1-1&keywords=data+analysis+using+regression+and+multilevel+hierarchical+models

Specifically written for Social Research, and presents example code for R and WinBugs

Seconding /u/khanable_ -- most of statistical theory is built on matrix algebra, especially regression. Entry-level textbooks usually use simulations to explain concepts because it's really the only way to get around assuming your audience knows linear algebra.

My Ph.D. program uses Casella and Berger as the main text for all intro classes. It's incredibly thorough, beginning with probability and providing rigorous proofs throughout, but you would need to be comfortable with linear algebra and at least the basic principles of real analysis. That said, this is THE book that I refer to whenever I have a question about statistical theory-- it's always on my desk.

This book is excellent:

https://www.amazon.com/Statistical-Rethinking-Bayesian-Examples-Chapman/dp/1482253445

http://www.amazon.com/Introduction-Probability-Edition-Dimitri-Bertsekas/dp/188652923X/ref=sr_1_1?ie=UTF8&qid=1394424420&sr=8-1&keywords=bertsekas+probability

You can find the video lectures from http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ or taking the course on edX https://www.edx.org/course/mitx/mitx-6-041x-introduction-probability-1296

*Solutions to the book exercises can be found on the book's website. Perfect for self-taught learner.

I would consider Jaynes to be one exception. His book Probability Theory: the Logic of Science is excellent.

Innumeracy

Excellent read.

Here you go! It's very helpful and has a wide range of topics so you can learn whatever you want. It uses Retrosheet, Lahman and Pitch Fx

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229/ref=sr_1_1?ie=UTF8&qid=1494296330&sr=8-1&keywords=analyzing+baseball+data+with+r

https://www.amazon.com/Analyzing-Baseball-Data-Chapman-Hall/dp/1466570229

This book covers everything related to how to get the data (Retrosheet, Lahman's, pitchf/x IIRC) and then how to do a lot of different stuff with R. It's a good place to start. You could probably find it cheaper than that Amazon link though.

Jaynes' Probability Theory is fantastic.

There are definitely ways to visualize algebraic concepts and many algebraic concepts crop up in geometry. Unfortunately, many books and classes won't emphasize visual intuition. So algebra may be harder for you. In some ways, you get over it even if it isn't your cup of tea, but there are also resources for transfering visual/geometric intuition onto algebraic concepts.

After reading it for myself, I recommend the books visual group theory by Nathan Carter, and algebra, concrete and abstract by Frederick Goodman. The first focuses a lot on visual intuition for group theory, but a lot concepts in group theory generalize to abstract algebra in general. The second book is a more traditional book, less focus on visual intuition, uses symmetry of geometrical objects and linear algebra for many of the examples.

Incorporating expert opinion into a Bayesian model is usually done through prior distributions instead of an additional feature. (As an aside, doing so is considered subjective Bayesian inference versus objective Bayesian inference).

As a quick overview, Bayesians usually make inference on the posterior distribution - a combination of the prior distribution (in your case, expert opinion), and the likelihood. As a really basic example, consider a setting where you have data on MI outcomes (no covariates at this point) - a series of 1's and 0's. A frequentist would likely take the mean of the data. As a Bayesian, you would consider this binomial likelihood and likely combine it with a beta prior. The default (non informative) prior would be to use a beta(1, 1) distribution. However, if in a prior dataset, you had observed four patients, three with an MI and one without, you could use a prior of beta(1+3, 1+1). See here for more details on beta-binomial.

In the above example, it's easy to incorporate prior information because we used a conjugate prior. While probably not exactly what you are doing for your dissertation, here's an overview of a conjugate prior with a linear regression from wikipedia. There are many more resources online for this that you can find by searching for something along the lines of "bayesian linear regression subjective conjugate prior". For a more detailed (introductory) overview of bayesian statistics, check out this book.

To be honest, as much as I'm a Bayesian, I think that creating an automatic model that incorporates expert opinion will be really difficult. Usually, subjective priors are chosen carefully, and there not always as interpretable as the beta-binomial posterior presented above. I think this goal is possible, but it would require a lot of though about how the prior is automatically constructed from a data set of surgeon's predictions. If you have any followup questions/would like more resources, let me know!

Edit: I guess I never really addressed the issue of predictive models. However, the difficult part will be constructing the prior automatically. If you can do this, predicting outcomes will be a simple change to make, especially in the case of linear model.

I enjoyed

Introduction to Probability Theory, Hoel et. al

Also,

Probability Theory, Jaynes

is essential. For probabilistic programming I would also look into

Bayesian Methods for Hackers

What is your background?

http://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126
Is a fairly standard first year grad textbook with I quite enjoy. Gives you a mathematical statistics foundation.

http://www.amazon.com/All-Statistics-Concise-Statistical-Inference/dp/1441923225/ref=sr_1_1?ie=UTF8&s=books&qid=1278495200&sr=1-1
I've heard recommended as an approachable overview.

http://www.amazon.com/Modern-Applied-Statistics-W-Venables/dp/1441930086/ref=sr_1_1?ie=UTF8&s=books&qid=1278495315&sr=1-1
Is a standard 'advanced' applied statistics textbook.

http://www.amazon.com/Weighing-Odds-Course-Probability-Statistics/dp/052100618X
Is non-standard but as a mathematician turned probabilist turned statistician I really enjoyed it.

http://www.amazon.com/Statistical-Models-Practice-David-Freedman/dp/0521743850/ref=pd_sim_b_1
Is a book which covers classical statistical models. There's an emphasis on checking model assumptions and seeing what happens when they fail.

You may also enjoy Probability Theory: The Logic of Science by E.T. Jaynes, and Information Theory, Inference and Learning Algorithms by David McKay.

If you want an extremely practical book to complement BDA3, try Statistical Rethinking.

It's got some of the clearest writing I've seen in a stats book, and there are some good R and STAN code examples.

This has been pretty much the standard textbook on Bayes
https://www.amazon.com/Bayesian-Analysis-Chapman-Statistical-Science/dp/1439840954/

I find Gilbert Strang's Introduction to Linear Algebra quite accessible, and seems to be aimed towards the practical (numerical) side of things. His video lectures are also quite good, IMHO.

Well is not exactly statistics, rather a bunch of anecdote on common mistakes and misconception about mathematics, but there is this book:

"Innumeracy: Mathematical Illiteracy and Its Consequences" by John Allen
(http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Consequences-Vintage/dp/0679726012)

and it's topic is vaguely related to OP's concern.

I haven't read it all but so far it was quite fun. Again is more anecdotal than scientific and the author might be a little condescending, but is worth reading.

This one? Damn, it's £40-ish. Any highlights or is it just a case of this book is the highlight?

It's on my wishlist anyway. Thanks.

Jaynes: Probability Theory. Perhaps 'rigorous' is not the first word I'd choose to describe it, but it certainly gives you a thorough understanding of what Bayesian methods actually mean.