Reddit reviews All of Statistics: A Concise Course in Statistical Inference (Springer Texts in Statistics)

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.

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• 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
  
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  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.
  
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I like Szekeres's A Course in Modern Mathematical Physics for referencing intro-grad-level material. It covers abstract linear algebra, differential geometry, measure theory, functional analysis, and Lie algebras, and teaches you some physics along the way.

More generally, the best "breadth" book on advanced mathematics is Princeton Companion to Mathematics by Gowers et al. and its slightly underachieving younger brother of a companion text, Princeton Companion to Applied Mathematics by Higham et al.. You won't properly learn advanced mathematics this way, but you'll get the bird's-eye view of modern research programs and the math underlying them.

If you want a more algebraic take on Szekeres's program to teach physicists all the math they need to know, check out Evan Chen's Napkin project, which is intended to introduce advanced undergrads (it's perfectly fine for grad students too) to a wide variety of advanced mathematics on the algebra side of things.

Since you're doing probability and statistics, check out Wasserman's All of Statistics and Knill's Probability Theory and Stochastic Processes for good, concise references for intro-grad-level material.

I will second what /u/Ovationification said, though. I didn't really learn anything with the above books, I just use them occasionally for reference or to think about pedagogy.

One of the standard recommendations for someone with a decent math background is All of Statistics by Wasserman. I personally found the style to be lacking on the pedagogical side in that there's next to no hand-holding when it comes to the exercises, but maybe you'll like it. The nice thing about it is that it covers much more than your usual "here's Bayes' theorem and a few things about sampling" book: bootstrapping, parametric inference, decision theory, causal inference, graphical models, some simulation methods, etc.

As for what next, it's hard to recommend anything without knowing exactly what you're interested in (biology is a pretty large field...).

Yes, think All of Statistics [https://www.amazon.com/All-Statistics-Statistical-Inference-Springer/dp/1441923225] is a really good starting point.

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.

Along the lines of probability, I recommend The Art of Probability. I also like the Schaum's Outlines of Probability and Statistics. If you want something more mathematical (calculus-based), All of Statistics by Wasserman is a solid reference.

The course itself is amazing. Probably because our teacher is great.

These are the topics we've covered so far:

1. Descriptive Statistics
   
   
2. Elements of Probability Theory
   
   
3. Inferential Statistics
   
   
4. Linear Regression
   
   Some recommended reading for this course:
   
   

• Larry Wasserman. All of Statistics: A Concise Course in Statistical Inference
  
  
• Charles M. Grinstead, J. Laurie Snell. Introduction to Probability
  
  
• Jackie Nicholas. Introduction to Descriptive Statistics

Hrmh, given your background I guess I would go with a suggestion of Wasserman for Statistical Inference or Casella and Berger which isn't really applied. If those are too much for you (which I doubt with your background), there is also Wackerly's Mathematical Statistics with Applications :)

sorry, Wasserman's all of statistics I'd suggest over Hogg and Craig given your direction. and yes, your roadmap will be a great bridge to getting into Bishop's and ESL, it'll get you where you're going (assuming the linear algebra/prob books are good, I'm not familiar with them).