Mathematical foundations of the calculus of probability
by J. Neveu · first published 1965
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…logarithm, etc. Now we are ready to do applied probability and statistics. And we have never mentioned either frequentism or Bayesianism. For more details, in an elegant presentation, see J. Neveu, Mathematical Foundations of the Calculus of Probability.
on “Frequentism and Bayesianism: A Practical Introduction” ↗…Theory, I and II, 4th Edition,\/} Springer-Verlag, New York.\ \ Kai Lai Chung, {\it A Course in Probability Theory, Second Edition,\/} ISBN 0-12-174650-X, Academic Press, New York.\ \ Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco.\ \ Neveu is succinct, gorgeous, but not easy. This material is NOT popular in US departments of mathematics. At Princeton, see Cinlar. Then should make some progress…
on “Ask HN: I want to learn statistics and data mining” ↗…conditioning (i.e., Bayesian) and, from there, in stochastic processes, Markov processes and martingales (astounding results). Books include L. Breiman, Probability, K. Chung, A Course in Probability Theory, J. Neveu, Mathematical Foundations of the Calculus of Probability. And there are more. IIRC Breiman and Neveu were both students of M. Loeve at Berkeley -- sure, can also get Loeve's Probability in two volumes. Of these, Neveu is my favorite; it's elegant; but for…
on “Why squared error? (2014)” ↗…used in statistics, but not defined in statistics. There is a polished, elegant, and thorough treatment of such independence, including of uncountably infinite sets of random variables, in J. Neveu, Mathematical Foundations of the Calculus of Probability.
on “Does superdeterminism save quantum mechanics?” ↗"Credibility"? Look up the Radon-Nikodym theorem. I suggest Neveu, 'Mathematical Foundations of the Calculus of Probability'. "Rant" is an INSULT. The 'rant' was in the article complaining that they could not get their paper published. I got them back on track.
…the values of such random variables, we can get some even integers selected independently. In good treatments of probability, e.g., (from my bibliography file, with TeX markup) Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco. independence is very nicely developed, e.g., to handle even a set of uncountably infinitely many independent random variables. As I recall, Neveu was a Loeve student.…
on “Probability (1963)” ↗…should try to find that distribution; instead, commonly in practice we have little hope of knowing much about the distribution! Or, one of the best books in probability I know of is Jacques Neveu, 'Mathematical Foundations of the Calculus of Probability', Holden-Day. and there is at most only meager mention of specific distributions in the whole book!
on “List of probability distributions” ↗…theory again, and more on functional analysis. Also von Neumann's cute proof of the Radon-Nikodym theorem. Nice treatment of Fourier theory. Some more nice material not easy to find elsewhere. Neveu, Mathematical Foundations of the Calculus of Probability A second or third book on probability. Succinct. Elegant. My candidate for the most carefully done, serious writing ever put on paper. Earl A. Coddington, An Introduction to Ordinary Differential…
on “What's a Mathematician to do?” ↗…Kept busy. Had a great time. Also got into computing in a fairly big way. Got some nice items, e.g., two new cars, etc. My favorite book on my bookshelf, including for applied math, is J. Neveu, Mathematical Foundations of the Calculus of Probability. Worked in industry and saw some problems in combinatorial optimization, deterministic optimal control, and stochastic optimal control, identified a problem in stochastic optimal control and found an…
on “Designing a Business Card in LaTeX” ↗…could write P(X in {1,2,3,4}) When the elementary notation is bad, a bit tough to take the more advanced parts seriously. A polished, elegant treatment of these basics is early in Jacques Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, 1965. Neveu was a student of M. Loeve at Berkeley, and can also see Loeve, Probability Theory, I and II, Springer-Verlag. A fellow student of Neveu at Berkeley under Loeve was…
on “Stanford Lecture Notes on Probabilistic Graphical Models” ↗…Mathematical Analysis. aka, "Baby Rudin". It's highly precise and challenging. For more, H. Royden, Real Analysis W. Rudin, Real and Complex Analysis L. Breiman, Probability M. Loeve, Probability J. Neveu, Mathematical Foundations of the Calculus of Probability The last two are challenging. For Bayesian, that's conditional expectation from the Radon-Nikodym theorem with a nice proof by John von Neumann in Rudin's Real and Complex Analysis. After those texts,…
on “Ask HN: What maths are critical to pursuing ML/AI?” ↗…or conditional independence, and there are some surprisingly good, intuitive approaches to these two. Next, it works. Now, call it done. 'Controversy' over with. For the math, see, e.g., J. Neveu, 'Mathematical Foundations of the Calculus of Probability'. 'Frequentist'? That's essentially answered by the weak/strong laws of large numbers. Bayesian? That's essentially conditional probability, with the advanced approach via the Radon-Nikodym theorem…
on “Modern Science and the Bayesian-Frequentist Controversy (2005) [pdf]” ↗Jacques Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco. Random variables and the associated probability theory and conditional probability theory are surprisingly relevant, especially now with computing, in the real world. That…
on “Ask HN: What books fundamentally changed the way you think about the world?” ↗…case of E[Y|X], conditional expectation. The detailed definition of E[Y|X] is based on the Radon-Nikodym theorem. W. Rudin's Real and Complex Analysis gives the novel von Neumann proof. Then in Neveu, Mathematical Foundations of the Calculus of Probability is elegant coverage of E[Y|X].
on “The Unreasonable Effectiveness of Conditional Probabilities” ↗…Walter Rudin, {\it Real and Complex Analysis\/} with Lebesgue integration and, then, Banach and Hilbert vector spaces. Walter Rudin, {\it Functional Analysis\/} with Fourier theory. Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability\/} with random variables, that is, functions from a probability space to, usually, the set of real numbers with convergence results, building on the work A. Kolmogorov building on the work of H. Lebesgue.
on “Functions Are Vectors (2023)” ↗