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I know people love to close duplicates so I am not asking for a reference to start learning statistics (as here).

I have a doctorate in mathematics but never learned statistics. What is the shortest route to the equivalent knowledge to a top notch BS statistics degree and how do I measure when I have achieved that.

If a list of books would suffice (assuming I do the exercises lets say), that's terrific. Yes, I expect working out problems to be an implicit part of learning it but I want to fast track as much as realistically possible. I am not looking for an insanely rigorous treatment unless that is part of what statistical majors generally learn.

amoeba
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John Robertson
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    what field of mathematics you received your doctorate? This might be relevant. – mpiktas Jan 25 '11 at 19:42
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    Could you share with us *why* you want to learn stats? Curiosity? Needed for a project or research? Wanting to change jobs? Need to teach some courses? Want to collaborate with statisticians as the theoretical person? – whuber Jan 25 '11 at 19:51
  • It would be useful in my work which involves a heavy amount of machine learning. It would also be useful at times as my (very small) company occasionally could use a genuine statistician. – John Robertson Jan 27 '11 at 00:58
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    I think it's almost always important to develop domain-specific expertise as well. A lot of statistics is learning the models relevant to specific fields. – Tristan Feb 22 '11 at 00:30
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    Try reversing "statistician wants equivalent knowledge to a quality maths degree" - there is not likely to be any fast routes. – probabilityislogic Aug 27 '13 at 10:12
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    Roll a dice 100 times and you are done. ;) – wolfies Nov 22 '17 at 11:37

9 Answers9

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(Very) short story

Long story short, in some sense, statistics is like any other technical field: There is no fast track.

Long story

Bachelor's degree programs in statistics are relatively rare in the U.S. One reason I believe this is true is that it is quite hard to pack all that is necessary to learn statistics well into an undergraduate curriculum. This holds particularly true at universities that have significant general-education requirements.

Developing the necessary skills (mathematical, computational, and intuitive) takes a lot of effort and time. Statistics can begin to be understood at a fairly decent "operational" level once the student has mastered calculus and a decent amount of linear and matrix algebra. However, any applied statistician knows that it is quite easy to find oneself in territory that doesn't conform to a cookie-cutter or recipe-based approach to statistics. To really understand what is going on beneath the surface requires as a prerequisite mathematical and, in today's world, computational maturity that are only really attainable in the later years of undergraduate training. This is one reason that true statistical training mostly starts at the M.S. level in the U.S. (India, with their dedicated ISI is a little different story. A similar argument might be made for some Canadian-based education. I'm not familiar enough with European-based or Russian-based undergraduate statistics education to have an informed opinion.)

Nearly any (interesting) job would require an M.S. level education and the really interesting (in my opinion) jobs essentially require a doctorate-level education.

Seeing as you have a doctorate in mathematics, though we don't know in what area, here are my suggestions for something closer to an M.S.-level education. I include some parenthetical remarks to explain the choices.

  1. D. Huff, How to Lie with Statistics. (Very quick, easy read. Shows many of the conceptual ideas and pitfalls, in particular, in presenting statistics to the layman.)
  2. Mood, Graybill, and Boes, Introduction to the Theory of Statistics, 3rd ed., 1974. (M.S.-level intro to theoretical statistics. You'll learn about sampling distributions, point estimation and hypothesis testing in a classical, frequentist framework. My opinion is that this is generally better, and a bit more advanced, than modern counterparts such as Casella & Berger or Rice.)
  3. Seber & Lee, Linear Regression Analysis, 2nd ed. (Lays out the theory behind point estimation and hypothesis testing for linear models, which is probably the most important topic to understand in applied statistics. Since you probably have a good linear algebra background, you should immediately be able to understand what is going on geometrically, which provides a lot of intuition. Also has good information related to assessment issues in model selection, departures from assumptions, prediction, and robust versions of linear models.)
  4. Hastie, Tibshirani, and Friedman, Elements of Statistical Learning, 2nd ed., 2009. (This book has a much more applied feeling than the last and broadly covers lots of modern machine-learning topics. The major contribution here is in providing statistical interpretations of many machine-learning ideas, which pays off particularly in quantifying uncertainty in such models. This is something that tends to go un(der)addressed in typical machine-learning books. Legally available for free here.)
  5. A. Agresti, Categorical Data Analysis, 2nd ed. (Good presentation of how to deal with discrete data in a statistical framework. Good theory and good practical examples. Perhaps on the traditional side in some respects.)
  6. Boyd & Vandenberghe, Convex Optimization. (Many of the most popular modern statistical estimation and hypothesis-testing problems can be formulated as convex optimization problems. This also goes for numerous machine-learning techniques, e.g., SVMs. Having a broader understanding and the ability to recognize such problems as convex programs is quite valuable, I think. Legally available for free here.)
  7. Efron & Tibshirani, An Introduction to the Bootstrap. (You ought to at least be familiar with the bootstrap and related techniques. For a textbook, it's a quick and easy read.)
  8. J. Liu, Monte Carlo Strategies in Scientific Computing or P. Glasserman, Monte Carlo Methods in Financial Engineering. (The latter sounds very directed to a particular application area, but I think it'll give a good overview and practical examples of all the most important techniques. Financial engineering applications have driven a fair amount of Monte Carlo research over the last decade or so.)
  9. E. Tufte, The Visual Display of Quantitative Information. (Good visualization and presentation of data is [highly] underrated, even by statisticians.)
  10. J. Tukey, Exploratory Data Analysis. (Standard. Oldie, but goodie. Some might say outdated, but still worth having a look at.)

Complements

Here are some other books, mostly of a little more advanced, theoretical and/or auxiliary nature, that are helpful.

  1. F. A. Graybill, Theory and Application of the Linear Model. (Old fashioned, terrible typesetting, but covers all the same ground of Seber & Lee, and more. I say old-fashioned because more modern treatments would probably tend to use the SVD to unify and simplify a lot of the techniques and proofs.)
  2. F. A. Graybill, Matrices with Applications in Statistics. (Companion text to the above. A wealth of good matrix algebra results useful to statistics here. Great desk reference.)
  3. Devroye, Gyorfi, and Lugosi, A Probabilistic Theory of Pattern Recognition. (Rigorous and theoretical text on quantifying performance in classification problems.)
  4. Brockwell & Davis, Time Series: Theory and Methods. (Classical time-series analysis. Theoretical treatment. For more applied ones, Box, Jenkins & Reinsel or Ruey Tsay's texts are decent.)
  5. Motwani and Raghavan, Randomized Algorithms. (Probabilistic methods and analysis for computational algorithms.)
  6. D. Williams, Probability and Martingales and/or R. Durrett, Probability: Theory and Examples. (In case you've seen measure theory, say, at the level of D. L. Cohn, but maybe not probability theory. Both are good for getting quickly up to speed if you already know measure theory.)
  7. F. Harrell, Regression Modeling Strategies. (Not as good as Elements of Statistical Learning [ESL], but has a different, and interesting, take on things. Covers more "traditional" applied statistics topics than does ESL and so worth knowing about, for sure.)

More Advanced (Doctorate-Level) Texts

  1. Lehmann and Casella, Theory of Point Estimation. (PhD-level treatment of point estimation. Part of the challenge of this book is reading it and figuring out what is a typo and what is not. When you see yourself recognizing them quickly, you'll know you understand. There's plenty of practice of this type in there, especially if you dive into the problems.)

  2. Lehmann and Romano, Testing Statistical Hypotheses. (PhD-level treatment of hypothesis testing. Not as many typos as TPE above.)

  3. A. van der Vaart, Asymptotic Statistics. (A beautiful book on the asymptotic theory of statistics with good hints on application areas. Not an applied book though. My only quibble is that some rather bizarre notation is used and details are at times brushed under the rug.)

Nick Cox
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cardinal
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    @cardinal, ex-Soviet universities have separate undergraduate statistics studies. In Vilnius University for example you can get bachelor degree in statistics. From what I see with the students I wholeheartedly agree that master or even doctorate-level education is needed for interesting jobs. – mpiktas Feb 22 '11 at 07:13
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    @cardinal, @mpiktas 4 years in BS + 2 yearts in MS + 4 years in PhD makes ten years to learn something interesting :) I would give $+\infty$ to this great answer if possible. Most books are new to me. – Dmitrij Celov Feb 22 '11 at 10:41
  • I had a look at Boyd & Vandenberghe, Convex Optimization as it appears to be available online for free, and I found it a bit concerning. It had no mention of any of the following phrases: "conjugate gradient" "BFGS" "Broyden"(of BFGS). Those are standard modern methods, so I am confused why they would not be at least mentioned. – John Salvatier Feb 22 '11 at 17:22
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    @John Salvatier, you're correct that those methods aren't covered in this text. Then again, this strikes me as more a matter of taste, particularly since the text's main focus is not on algorithms. To wit, your concerns are directly addressed by the authors in the introduction (pg. 13). – cardinal Feb 22 '11 at 17:49
  • @John, My personal interpretation of the intent of this text is to provide the machinery necessary to *recognize* a problem as falling within the class of convex optimization problems rather than develop the algorithmic tools necessary to solve them (though some of this *is* presented) safely. Perhaps an appropriate analogy can be found in the standard treatments of linear regression. – cardinal Feb 22 '11 at 17:56
  • I would also recommend Boyd and Vandenberghe, especially if it is available for free. – Dikran Marsupial Feb 24 '11 at 11:27
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    @cardinal: Scandinavian universities usually do offer bachelors level degrees as well. That being said, I think statisticians take themselves a bit too seriously. I disagree that you'd need a doctorate degree in order to get an "interesting" job. I believe that as science and research becomes more and more cross-disciplinary statistics have been imposed on studies from many different areas. Half of the articles on high impact journals have some questionable statistical analysis, just to meet the demands, even though it might not make any sense, given in original context/domain of the problem. – posdef Feb 24 '11 at 12:52
  • [continuation from previous comment] ... I dont know if this is the case with the OP, but I can definitely relate to the dilemma of trying to fast forward, especially if you are done with your education. It's simply unrealistic to go through a long list of books, exercises and spend months and years to be able to do your job, which might not have anything to do with theoretical statistics... (I'm sorry if I sound too frustrated) – posdef Feb 24 '11 at 12:53
  • @posdef, I certainly empathize with your feelings of frustration. I take a little issue with some remarks that you've made that I suspect are mostly a product of this frustration. To respond to your comments completely would require an essay. Modern applied statistics is a mathematically and computationally sophisticated field which requires a fairly high degree of specialized knowledge. It strikes me as a bit presumptuous to want to become good/competent in the field without putting in the effort. Click on the link I provided for a more eloquent version of this. – cardinal Feb 24 '11 at 13:57
  • As someone who originally came from another discipline, I understand the desire to learn quickly enough to be productive. Unfortunately, my experience is that that is difficult and potentially dangerous. What is your background? If someone with a B.S. In statistics came to you and said "I need to know your field. Make me an expert as quickly as possible," what would you realistically suggest to them? When would *you* feel safe calling them an expert and handing them their own project that *your* business or livelihood depended on? – cardinal Feb 24 '11 at 14:03
  • Like I said, I think a well-thought-out essay is needed and this is hardly a good format for it, unfortunately. The points you raise regarding poor statistical analysis in the literature are valid. My argument would be that this is often due to precisely the problem we are trying to address in this thread. There are many, *many* people doing statistics without fully grasping what they are doing. There are also too few statisticians that have sufficient domain expertise in the area of application they are working in. – cardinal Feb 24 '11 at 14:07
  • @cardinal: While I mostly agree with you, what I find highly troubling is the assumption of "interest to master" the field of statistics. When I ask, or answer, a question on StackOverflow, I don't make assumptions on intentions to master anything, I only intend to learn to solve my problems. Likewise I don't assume people want to master the field of programming when they ask something there. Mastery, as you have mentioned, comes over time and effort of course. My experience SO vs CV and statistics vs programming have been bitterly different in terms of help/utility :( – posdef Feb 24 '11 at 15:22
  • @posdef, thanks for your comments. My response was tailored to the question of the OP who "wanted the equivalent knowledge of a quality stats degree." I read that as a desire for a certain degree of mastery. While programming requires a very significant degree of both logic and creativity, I would speculate that most of the questions on SO are about more direct implementation issues. CV is more geared towards helping others understand and apply statistics, which generally requires a deeper dialogue with the OP to do well (correctly) and involves addressing more subtleties. My $0.02. – cardinal Feb 24 '11 at 15:32
  • @cardinal: ah, i guess you're right... I'm sorry for creating an off-topic discussion like this, and for being very bitter on the subject. I gotta work on that, somehow. – posdef Feb 24 '11 at 15:48
  • @posdef, no worries. The discussion is interesting (to me) nonetheless. The best way to fight the feelings of frustration is to continue to learn when the opportunity arises. I, for one, am glad to have your participation here. Applied statistics as a field will only continue to grow through positive interactions with those interested in its applications. I would argue that in many cases the reverse is true as well. – cardinal Feb 26 '11 at 17:15
  • @cardinal: thanks! I'm trying to get rid of the prejudice and the frustration by trying to ignore the negative sides, and try to figure out the right "mindset". I believe science is somewhat like language, you'll never be fluent unless you learn to think in that language instead of thinking in some other language and translating in your head. We'll see how well that works out for me... :) – posdef Feb 28 '11 at 08:55
  • @cardinal: ignoring any quibbling over one text vs. another, the only thing I see in your list that is a serious omission is the lack of a reference on robust statistics. Maronna, Martin, and Yohai is a decent choice, or you can go 'classic' and put Huber or Hampel in there. – Wesley Burr Jun 05 '11 at 23:03
  • @Wesley: Thanks for the comment. Yes, explicit mention of robust statistics is missing. However, I haven't neglected it completely. Section 3.13 of Seber & Lee gives a nice 15-page introduction and several chapters of van der Vaart also deal with robust statistics. So, it's in there, if in slightly disguised form. Huber (1981) is a delightful little book, but I think it'd probably have to go under the "advanced" category in my list above. – cardinal Jun 09 '11 at 17:54
  • @cardinal: You make many good points. Two experiences I might add, once I had summer student who had just finished a Phd in math and wanted to switch to statistics - they had a very hard time getting that statistics was just not identifying functions and then doing calculations. Another time I had clients who had Phds in biophsyics with a lot of math. They were the least affected by any advice I offered... – phaneron Nov 23 '12 at 14:48
  • Agresti's categorical book is now in its 3rd edition. – Nick Cox Dec 20 '16 at 19:57
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    @cardinal Mood book was a great suggestion because nowadays it is difficult to find an introductory book on statistics that is formal enough for someone with a math background. Has anybody read this book new book? Panateros, "Statistics for Mathematicians" https://www.springer.com/us/book/9783319283395 – Dr Fabio Gori Nov 12 '18 at 14:30
  • @cardinal I have a PhD in Functional analysis, particularly dealing with Banach space theory. I am looking to learn more about time series forecasting models such as ARIMA, GARCH, etc. I particularly like Brockwell and Davis's Theory and Methods book as they really provide rigour. Do you happen to know any other books that are similar levels to Borckwell and Davis? – Idonknow Jul 22 '19 at 11:13
  • Is Convex Optimisation that useful for statisticians to know? – Noppawee Apichonpongpan Jun 22 '21 at 05:58
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I can't speak for the more rigorous schools, but I am doing a B.S. in General Statistics (the most rigorous at my school) at University of California, Davis, and there is a fairly heavy amount of reliance on rigor and derivation. A doctorate in math will be helpful, insomuch as you will have a very strong background in real analysis and linear algebra--useful skills in statistics. My statistics program has about 50% of the coursework going to support the fundamentals (linear algebra, real analysis, calculus, probability, estimation), and the other 50% goes towards specialized topics that rely on the fundamentals (nonparametrics, computation, ANOVA/Regression, time series, Bayesian analysis).
Once you get the fundamentals, jumping to the specifics is usually not too difficult. Most of the individuals in my classes struggle with the proofs and real analysis, and easily grasp the statistical concepts, so coming from a math background will most definitely help. That being said, the following two texts have pretty good coverage of many topics covered in statistics. Both were recommended in the link you provided, by the way, so I wouldn't say your question and the one you linked are necessarily uncorrelated.

Mathematical Methods of Statistics, by Harald Cramer

All of Statistics: A Concise Course in Statistical Inference, by Larry Wasserman

Christopher Aden
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    +1 All of Statistics: it would be a great place to start. – Simon Byrne Feb 21 '11 at 23:20
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    the UC-Davis program looks good and I think you'll get a great education there. I wouldn't consider it "less rigorous" than other places. I thought the comment on their "integrated B.S./M.S. degree" **[page](http://anson.ucdavis.edu/undergrad/bs-ms)** was interesting and relevant to the thread: "There is a high demand for statisticians, but the knowledge and skill achieved by those with a BS degree in Statistics are often not sufficient for the needs in the [government or industrial] workplace." – cardinal Feb 26 '11 at 23:57
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The Royal Statistical Society in the UK offers the Graduate Diploma in Statistics, which is at the level of a good Bachelor's degree. A syllabus, reading list, & past papers are available from their website. I've known mathematicians use it to get up to speed in Statistics. Taking the exams (officially, or in the comfort of your own study) could be a useful way to measure when you're there.

Scortchi - Reinstate Monica
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    The Graduate Diploma exams are effectively final year undergraduate exams; for "staging" purposes there are lower level certificates that can be taken first. The RSS exams are available, if I recall correctly, worldwide with the exception of Hong Kong (which has its own statistical society and exams). An alternative is the undergraduate Diploma in Statistics offered by distance learning by the Open University in the UK, but again available worldwide. This is of slightly lower level than the RSS Grad Dip so may be seen as preparation for it. As a taught course it's substantially more expensive. – Silverfish Nov 05 '13 at 03:28
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I would go to the curriculum websites of the top stats schools, write down the books they use in their undergrad courses, see which ones are highly rated on Amazon, and order them at your public/university library.

Some schools to consider:

Supplement the texts with the various lecture video sites such as MIT OCW and videolectures.net.

Caltech doesn't have an undergrad degree in statistics, but you won't go wrong by following the curriculum of their undergrad stats courses.

Neil McGuigan
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    this seems like a bit of an odd list. To my knowledge, Carnegie Mellon is the *only* school on that list that (formally) offers an undergraduate degree in statistics. Neither Caltech nor MIT even have graduate programs in statistics. – cardinal Feb 22 '11 at 00:07
  • @cardinal. why must you doubt me? :) I put in links to the undergrad stats courses at those fine institutions. Also, mixing and matching courses from the best schools will beat out following a degree path from a worse school. – Neil McGuigan Feb 22 '11 at 00:28
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    OCW is certainly a very fine resource and a great initiative. This is no doubting that. As for your assertion that mixing and matching from the "best schools" is a superior solution, I find that highly suspect, particularly for undergraduate studies. While a highly motivated student is bound to get a very good undergraduate education at any of those schools, an undergrad education as good or better can be found at many, many "worse" schools. Schools such as those that you list do tend to "win out" for graduate education, I would say. – cardinal Feb 22 '11 at 01:40
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    Actually, this was the first thing I tried. I tried this before posting the question. Finding a list of courses wasn't hard, but finding information about which books actually got used for those courses and what sections of those books were covered was much more difficult. – John Robertson Feb 24 '11 at 21:02
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Regarding the measurement of your knowledge: You could attend some data mining / data analysis competitions, such as 1, 2, 3, 4, and see how you score compared to others.

There are a lot of pointers to textbooks on mathematical statistics in the answers. I would like to add as relevant topics:

  • the empirical social research component, which comprise sampling theory, socio-demographic and regional standards
  • data management, which includes knowlegde on databases (writing SQL queries, common database schemes)
  • communication, how to present results in a way the audience stays awake (visualization methods)

Disclaimer: I am not a statistician, this are just my 2cents

Karsten W.
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I come from a computer science background focusing on machine learning. However, I really started to understand (and more important to apply) statistics after taking a Pattern Recognition course using Bishop's Book https://www.microsoft.com/en-us/research/people/cmbishop/#!prml-book

here are some course slides from MIT:
http://www.ai.mit.edu/courses/6.867-f03/lectures.html

This will just give you the background (+ some matlab code) to use statistics for real work problems and is definitely more on the applied side.

Yet, it highly depends on what you want to do with your knowledge. To get a measure for how good you are you might want to browse the open course ware of some university for advanced statistics courses, to check if you know the topics covered. Just my 5 cent.

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kgarten
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E.T. Jaynes "Probability Theory: The Logic of Science: Principles and Elementary Applications Vol 1", Cambridge University Press, 2003 is pretty much a must-read for the Bayesian side of statistics, at about the right level. I'm looking forward to recommendations for the frequentist side of things (I have loads of monographs, but very few good general texts).

Scortchi - Reinstate Monica
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Dikran Marsupial
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    I would suggest It's a must read for *anybody* who wants to be a good statistician, Frequentist, Bayesian or anything else. – probabilityislogic Jan 26 '11 at 00:16
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    I disagree, Jaynes' book is a terrible recommendation in this circumstance: 1) the notation is sloppy and non-standard, which makes it difficult to cross reference with other sources, 2) he's long winded and gets bogged down in silly and irrelevant arguments (the OP asked for the "shortest route") 3) there's also the errors (such as the marginalization paradox) – Simon Byrne Feb 21 '11 at 22:56
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    @Dikran Marsupial, do you own the Schervish text on statistical inference? I've been on the fence regarding whether to purchase it or not, so was curious, since you appear to align yourself pretty strongly with the Bayesian approach. – cardinal Feb 22 '11 at 19:07
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    I wouldn't say I was strongly aligned to the Bayesian approach. It is the approach I understand the best, which is not the same thing. Essentially I am an engineer at heart, and I want both tools in my toolbox, maintained in good order! A proper understanding of the benefits and disadvantages of each approach is what we should aim for. I haven't got Shervishes book, but I did read a paper of his on Bayes factors that seemed quite flawed to me (I'll see if I can find it and post a question for someone to explain it to me!). – Dikran Marsupial Feb 22 '11 at 19:24
  • @Dikran, Your (potential) question sounds interesting. I look forward to a post on it. – cardinal Feb 24 '11 at 00:56
  • I expect I am just missing something, but it is the paper by Lavine and Schervish "Bayes factors: what they are and what they are not". The example 2 seems a bit odd as they use a Bayes factor in a situation where they have a well specified (and highly informative) prior on the hypotheses. As a result it seems unsurprising that it behaves a little oddly given that the information (mu) is used in computing the Bayes factor, but then treated as unknown in the hypothesis test itself. I like reading papers on what things actually mean though and being wrong is a good way of learning! – Dikran Marsupial Feb 24 '11 at 11:25
  • Late in the day, but I added a plus to the comment against Jaynes. It's a book of some considerable interest to those who have their own slant on statistics already. As an introduction it could only be singularly bizarre, even for the mathematically well educated. Cambridge U.P. were completely right to publish it, flaws and all, but the flaws reduce its pedagogic value considerably. – Nick Cox Nov 26 '15 at 15:52
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I have seen Statistical Inference, by Silvey, used by mathematicians who needed some workaday grasp of statistics. It's a small book, and should by rights be cheap. Looking at http://www.amazon.com/Statistical-Inference-Monographs-Statistics-Probability/dp/0412138204/ref=sr_1_1?ie=UTF8&s=books&qid=1298750064&sr=1-1, it seems to be cheap second hand.

It's old and concentrates on classical statistics. While it's not highly abstract, it is intended for a reasonably mathematical audience - many of the exercises are from the Cambridge (UK) Diploma in Mathematical Statistics, which is basically an MSc.

mcdowella
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I think Stanford provides the best resources when it comes to flexibility. They even have a machine learning course online that would provide you with a respectable base of knowledge when it comes to designing algorithms in R. Search it up on Google and it will redirect you to their Lagunita page where they have some interesting courses, most of them being free. I have Tibshirani's books, Introduction to Statistical Learning' and 'Elements of Statistical Learning' in PDF formats and both are extremely good resources.

Since you're a mathematician, I would still advise you to not fast track as that wouldn't provide you with a solid base that you might find very helpful in the future if at all you start doing some serious machine learning. Treat statistics as a branch of mathematics for getting insights from data, and that requires some work. Other than that, there are tons of online resources, Johns Hopkins provides similar stuff as Stanford. Although experience always pays, a respectable credential will always reinforce that base. You can also think of the specific fields that you would like to enter; by that I mean whether you want to go into text analytics or applying your math and stats skills in finance. I come in the latter category so I have a degree in econometrics where we studied finance + statistics. A combination can always be very good.

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