Probability
by Leo Breiman · first published 1968
71% recommend29% neutral
Why HN recommends it
Rigorous measure-theoretic probability text, very well written. Useful for working through foundations and offers a treatment of ergodic theory. A standard graduate reference.
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…measurability. Part of the reason is the issue of 'measurable selection': This can be a deep subject, but often a relatively simple way out is via regular conditional probabilities as in Leo Breiman, 'Probability', ISBN 0-89871-296-3, SIAM, Philadelphia. For David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and Sons, Inc., New York. this has likely the easiest mathematical treatment of…
on “The pitfalls of jargon and the damage it does to mathematics” ↗…of Operations Research. Neveu was a student of M. Loeve at Berkeley. So was the current darling of machine learning, L. Breiman, because of his Classification and Regression Trees (CART). Breiman's Probability as published by SIAM is generally easier reading than Neveu. For stochastic processes, there are several relatively different directions to go. Martingale theory is gorgeous, astounding, amazing, with…
on “Berkeley offers its data science course online for free” ↗Hehe ok —- I also love Breiman’s Probability book. It’s really a standout on Ergodic theory. And Breiman et al.’s book on Trees is surprisingly rich, talking about all sorts of stuff besides trees.
on “Foundations Machine Learning” ↗…building on top of the axiomatic foundation that you outlined for decades. This is old news. Yup, essentially since Kolmogorov's 1933 paper. The axiomatic approach I gave is just what is used in Breiman's Probability, as published by SIAM. It's a super nicely written book. I have never seen his book on statistics.
on “Frequentism and Bayesianism: A Practical Introduction” ↗…And we should note that much of the AI interest in regression is based on the work of L. Breiman in Classification and Regression Trees (CART). Breiman was, IIRC "an academic probabilist"; his text Probability (one of my favorites, e.g., for measurable selection) was all based heavily on measure theory; and his work on CART started by trying to get fits and predictive models starting with complicated data from…
on “Carnegie Mellon Launches Undergraduate Degree in Artificial Intelligence” ↗…Analysis: A Modern Approach to Advanced Calculus,\/} where don't take the proofs very seriously but to see how physical science and engineering look at calculus of several variables. Neveu is a great last probability text but not a good first text. So, before Neveu, look quickly, not very seriously, at whatever, including in some introductory statistics texts. Also, Breiman's Probability is easier to read than Neveu.…
on “Ask HN: What are your favorite statistics and probability textbooks?” ↗For ergodic theory, there were a few really good lectures, and I have good notes, in a course I took from a star student of E. Cinlar. For Breiman Probability: (1) I liked his start, nice, simple, intuitive, on how the law of large numbers works. Of course the super nice proof is from the martingale convergence theorem, but Breiman's start is nice. (2) His…
on “Foundations Machine Learning” ↗…Optimization -- linear programming, linear programming on networks (often get integer programming for free -- looks NP-complete but it's not), nonlinear programming (especially the Kuhn-Tucker conditions). Probability based on measure theory -- Breiman, Chung, Neveu, Loeve, Chow and Teicher. The above is a bit minimal; more could be from important up to really valuable. Generally you are trying to find tools in math…
on “Manuel Blum: Advice to a beginning graduate student” ↗…university department should help. E.g., if you work through Halmos 'Finite Dimensional Vector Spaces', Rudin's 'Principles of Mathematical Analysis', MacLane and Birkhoff's 'Algebra', or Leo Breiman's 'Probability', along with supplementary sources for each, then maybe ask to sit in, totally informally, just to observe, one or a few sessions of a corresponding university course. And get some homework assignments…
on “Ask HN: Learn Math the Hard Way” ↗…frequentist of Bayesianism. That should be welcome good news. People have been building on top of the axiomatic foundation that you outlined for decades. This is old news. You mention Breiman's book on Probability. He has also a book on Statistics. Maybe there is something to it after all?
on “Frequentism and Bayesianism: A Practical Introduction” ↗…SAT scores, etc. Let the children get through the Beethoven piano sonatas and the Bach unaccompanied violin pieces, Rudin's 'Principles of Mathematical Analysis', Royden's 'Real Analysis', Breiman's 'Probability', and the 'Feynman Lectures on Physics' -- should keep them busy for a while! What do you want? Lots of really healthy grandchildren or someone to sweep out the garage?
on “Don't dumb girls down” ↗