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Foundations of probability theory
31–40 of 54 posts
Re: Foundations of probability theory
#32The fact that Shannon entropy is still relevant in numerous modern mathematics research is amazing. Apart from minor typos that make him a "sloppy mathematician", he is a good educator based on my personal experience taking grad. courses from him. He's not the passionate high school STEM teacher type, but he offers great insights. I think which textbook he used is somewhat secondary. For courses he had taught before,…
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Re: Foundations of probability theory
#33I don't know what mathematicians think of it, but I enjoyed "Probability Theory: The Logic of Science". https://www.google.com/#q=probability+the+logic+of+science
It's interesting that starting from the principles he states in section 1.7 the rules of probability can be derived. This formulation is from http://arxiv.org/abs/1507.06597 (where it is shown that the measure-theoretical approach is just an special case):
(Representation) Degrees of plausibility pl(A|B) can be identified as unique real numbers p ∈ R; and
(Qualitative Correspondence With Common Sense; Logical Internal Consistency) This has three sub-Principles:
– If there is more than one path to a correct conclusion, all such paths must lead to the same plausibility result;
– In assessing pl(A|B), You must always use all of the available information that You regard as relevant to the assessment; and
– Equivalent states of information about (A|B) always lead to the same pl(A|B).
Re: Foundations of probability theory
#34Earlier quoted context omitted.
This is possible, and in fact probably implemented in some probabilistic programming languages, but I think you are looking at the wrong direction. The point is that even for fairly simple real use cases, the computation complexity is so huge, that all computers in the world couldn't compute it in your lifetime if you don't employ some approximation or optimization and stick to naive algorithms. So, that is what the…
All those probabilistic programming languages will become exponentially faster once we have feasible quantum computers, since BPP \in BQP. We currently use a weaker inclusion, BPP \in PSPACE, as the core execution model.
Check your facts, because this is simply not true.
Re: Foundations of probability theory
#35I don't know what mathematicians think of it, but I enjoyed "Probability Theory: The Logic of Science". https://www.google.com/#q=probability+the+logic+of+science
I enjoy it, his papers and that books sit proudly on my desk. It's interesting that his collection of papers gives an entirely different approach to his approach to probability theory and hardly focuses on it being an extension of Aristotelean logic.
Re: Foundations of probability theory
#36I have a question that might be related to the topic. I have found that often, when solving problems related to probability theory, it is more convenient to think in terms of combinations than to think in terms of probabilities (perhaps because I'm a programmer). Would it be possible to devise a programming language that allows me to program a filter that selects the desired outcomes out of the complete set of combin…
I think that would only work in the discrete case or for simple problems, however something relevant
>“Information theory must precede probability theory and not be based on it.” A.N.Kolmogorov, in [Kolmogorov, 19831.
and it was a combinatorics paper
Re: Foundations of probability theory
#37Looked at the text to be used, Durrett. There's more and with higher quality in any of M. Loeve, J. Neveu, K. Chung, and L. Breiman. For what Tao wrote on his page about determinism or whatever on that page -- just f'get about that. And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
To each their own. Durrett's text is perfectly fine. The books you mention are older, and perhaps you originally learned from them, so you still hold a torch?
In particular, sending people to Loeve for their first course in measure theoretic probability would be really cruel.
Personally, I learned from Feller, and Billingsley, and finally Durrett.
Re: Foundations of probability theory
#38Looked at the text to be used, Durrett. There's more and with higher quality in any of M. Loeve, J. Neveu, K. Chung, and L. Breiman. For what Tao wrote on his page about determinism or whatever on that page -- just f'get about that. And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
To each their own. Durrett's text is perfectly fine. The books you mention are older, and perhaps you originally learned from them, so you still hold a torch? In particular, sending people to Loeve for their first course in measure theoretic probability would be really cruel. Personally, I learned from Feller, and Billingsley, and finally Durrett.
> To each their own. Durrett's text is perfectly fine. The books you mention are older, and perhaps you originally learned from them, so you still hold a torch?
Durrett seems on most of the topics to have less than any of the four texts I mentioned.
> In particular, sending people to Loeve for their first course in measure theoretic probability would be really cruel.
I listed the authors in reverse order in which to read them! The easiest start is the last, Breiman. But some things in Loeve are good, e.g., sufficient statistics. Neveu is my favorite as the most elegant, but his exercises are the most difficult. Breiman and Chung are fine. IIRC, both Breiman and Neveu were Loeve students.
> Personally, I learned from Feller, and Billingsley, and finally Durrett.
I used Feller I and/or II for reference. Otherwise his writing seemed to lack an overall unification.
The only Billingsley text I used, and then only for a small topic, was Convergence of Probability Measures.
Re: Foundations of probability theory
#39Earlier quoted context omitted.
And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day. Could you expand on this? I'm reading his post now, but from the outside, it seems unlikely that Terry got the concepts completely wrong, and more likely that he's using slightly different definitions for the concepts than you are expecting.
No, he's just being sloppy. That can be a real pain for students trying to learn. But, be warned: A lot of people work with probability, but only a tiny fraction ever had a course in graduate probability . So, eventually have to learn to put up with, and sometimes rewrite, some of what is written that is not very precise. Here are the accepted definitions: Take a non-empty set, usually denoted by Omega, and call it t…
>the only kind should bother with anyway!
I hope all this studying will pay off then, I have fremlin vol1/vol2 sitting underneath my text while I handle the per-requisite material hopefully the journey will pay off.
Measure theory can be used with a lot of stuff I guess.
Re: Foundations of probability theory
#40I don't know what mathematicians think of it, but I enjoyed "Probability Theory: The Logic of Science". https://www.google.com/#q=probability+the+logic+of+science
Its take on probability as a way of modelling our brain is refreshing, for instance, the beginning of chapter 1: Suppose some dark night a policeman walks down a street, apparently deserted; but suddenly he hears a burglar alarm, looks across the street, and sees a jewelry store with a broken window. Then a gentleman wearing a mask comes crawling out through the broken window, carrying a bag which turns out to be ful…
Ah, but policeman's actual thought process might have actually been purely deductive or purely reflexive, it might only be that the probabilities justify a series of a discreet calculations.