stackexchange, please suggest me some books on probability & statistics that are unconventional in their approach to these subjects.

I think it is better to describe what i mean by "unconventionality" by example.

Some of this kind of books are

  1. "Probability Theory: The Logic of Science" by E. T. Jaynes. The famous book where the author uses Cox's theorems for the laws of probability and interprets probability as an extension of logic.
  2. "Probability and Finance: It's Only a Game!" by Shafer and Vovk. In this book the authors derive the laws of probability from game theory.
  3. "Probability via Expectation" by Peter Whittle. The author develops the theory of probability from axioms on the expectation functional rather than on probability measure.
  4. "Radically Elementary Probability Theory" by Edward Nelson. The author uses nonstandart analysis in his treatment of probability.
  • $\begingroup$ A clarification? Maybe it's not incorrect that Shafer and Vovk "derive the laws [i.e. axioms?] of probability from game theory", but I think it gives the wrong impression. What Shafer and Vovk think is important about their books is that they contain proofs of standard limit theorems and other results using certain sorts of abstract repeated games involving betting and prediction. Shafer and Vovk argue there are benefits to proving these theorems without measure-theoretic probability axioms, and that their approach is more general, covering cases standard probability theory doesn't address. $\endgroup$ – Mars Jul 23 at 16:26

How to Gamble If You Must, Lester E. Dubins & Leonard J. Savage

This classic of advanced statistics is geared toward graduate-level readers and uses the concepts of gambling to develop important ideas in probability theory.

  • $\begingroup$ I remember it because it has a great title. $\endgroup$ – GEdgar Sep 25 '16 at 17:09
  • $\begingroup$ I'll bet the OP gets this. $\endgroup$ – Daniel Buck Sep 25 '16 at 17:17
  • 2
    $\begingroup$ As I recall, one unconventional feature of Dubins & Savage is that finite additivity is assumed, but countable additivity is not. So this approach is a departure from the Kolmogorov Axioms. $\endgroup$ – BruceET Sep 25 '16 at 20:39
  1. Imprecise probability books:

Statistical Reasoning with Imprecise Probabilities, by Peter Walley (1990).

Lower Previsions by Troffaes and de Cooman (2014).

Introduction to Imprecise Probabilities edited by Augstin, Coolen, de Cooman, and Troffaes (2014).

Those three books are more or less on the same subject: Imprecise probability, mostly from a generalized Bayesian perspective. Walley is the classic text on the subject, but it's also worth mentioning Isaac Levi's 1980 philosophical work The Enterprise of Knowledge. For me, personally, the first few chapters of Lower Previsions (i.e. lower expectations) was a better introduction to the area than the first few chapters of Introduction, but everyone's different. There is material in the Introduction that I found helpful while reading Lower Previsions; their initial chapters are in some respects complementary. In addition, Introduction includes introductory survey chapters on a wide variety of topics, such as imprecise stochastic processes, that aren't covered in an introductory manner anywhere else. (Introduction also includes a chapter by Shafer and Vovk that introduces their game-theoretic probability approach in a somewhat different manner than in their books.)

  1. Algorithmic information theory books. Maybe these don't count as "unconventional", since they are part of an established field. However, these books contain material on probability and randomness that's foundational in a way that's very different from most foundational work on probability. Among other things, this area connects to the idea of statistical testing via e.g. Martin Löf's work on randomness and other ideas covered in in these books.

An Introduction to Kolmogorov Complexity and Its Applications by Li and Vitanyi. Classic textbook. I have found the third edition very useful, but difficult to understand at times because of typos, an attempt to avoid parts of mainstream probability theory sometimes, or lack of back-references to material introduced much earlier in the book. A fourth edition just came out, though, so that might avoid some of the drawbacks.

Information and Randomness by Calude. Not as much hand-holding as Li and Vitanyi, but maybe that's better given how L & V do it sometimes.

Algorithmic Information Theory by Chaitin. Quirky but interesting perspectives.

  1. More Shafer and Vovk:

They have a new 2019 book on the same topic as the book mentioned in the question. Game-Theoretic Foundations for Probability and Finance contains new results, but less of the history, philosophy, and initial hand-holding of the the book mentioned in OP's question. Up to you which is a better starting point.


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