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> I notice you didn't propose a way to choose the sample size for a randomized trial in your reply. We're not communicating well. To respond to your question

by graycat 12y ago

> I notice you didn't propose a way to choose the sample size for a randomized trial in your reply. We're not communicating well. To respond to your question about sample size, I'd have to look into some of the details of your question. And I can say now, that question, those details, and any answer all have essentially nothing to do with the axiomatic foundation of probability I gave; the answer is the same or essentially so independent of those foundations down in the deep sub basement of the subject. Whether something is "only probability" or also "stats" can be important in practice -- e.g., will find relatively little about a lot of important work in statistics in Neveu's book on probability. And there is a lot of practical knowledge in applied statistics, e.g., how well principle components analysis tends to work in practice (quite well). E.g., there is a lot in survey and sampling techniques. And a lot that is in mathematical statistics has yet to have been derived as fully clean applied mathematics from only something like Neveu. Still, basically mathematical statistics is applied probability which in principle can all be done back to Neveu and only a little more, e.g., matrix theory, some combinatorics, maybe some group theory for bootstrap and resampling plans, etc. Again, my post was on the foundations of probability and showing that there we did not need to mention either frequentism or Bayesianism. That is, maybe my post would be helpful for people struggling with frequentism or Bayesianism -- I'm saying, since 1933, for just the foundations, get to f'get about both of them. I studied probability from a star student of Cinlar at Princeton and from books by Neveu, Chung, Breiman, Loeve, and others. I did a lot in applied probability and applied statistics. I've published in mathematical statistics. My Ph.D. dissertation was in applied probability. So? I have background enough to post. But here I only wanted to explain the foundations of probability and not compete with anyone or compare my expertise with anyone. Instead, I'm just reporting some news current as of 1933. It does appear to me that now for nearly all serious work in probability and stochastic processes, the Kolmogorov foundations are nearly universally accepted; thus, readers are safe in taking seriously the news I reported. Maybe someday I will return to pure and applied probability, but for now my interests are in my startup. There, now, mostly the work is in software and other parts of business. At the core, my startup is some work in applied probability, but I did that months ago and long since have had the corresponding computations in solid software. So, for now, more background in probability is not on my TODO list. Again, here I'm just giving some HN readers the news that as of 1933 get to f'get about the frequentism and Bayesianism foundations of probability.

kgwgk 12y ago
Leaving aside that Kolmogorov's measure-theoretic approach is not the only axiomatic definition of probability (Cox's axioms yield a quite similar foundation, though with finite additivity only), you won't find anyone here that says there is a problem with the mathematical construction. The frequentist/Bayesian debate is related to the INTERPRETATION of probability. Kolmogorov won't help you to map the real world to the probability space. Let's say we have a loaded coin, we want to estimate the probability of getting tails (assume this is a i.i.d. random variable). Alice decides to keep throwing until she gets a tail: she gets the sequence HHT Bob decides to throw the coin three times: he gets the sequence HHT Alice takes her event A, her sigma-algebra, the whole shebang, and produces an interval estimate for p. Bob takes his event A (which happened to be the same) and his probability space (which is different because the experimental design is different), and produces a different estimate for p. If you think that getting different results from the same data makes sense, you might be a frequentist. If you think that it doesn't, you might be a Bayesian. If you think that the question is not relevant because you can't derive the answer from your axioms, you might at least understand what we're talking about.
- graycat 12y ago
  This is all mixed up. Alice has real random variables X_1 (borrowing TeX notation for a subscript), X_2, X_3. We assume that {X_i|i = 1, 2, 3} is independent. We assume that for some number p in [0,1], P(X_i = 0) = p -- Alice uses 0 for H and 1 for T. Alice observes that X_1 = 0, X_2 = 0, and X_3 = 1. Now, for the set of real numbers, R, Alice has a Borel measurable function f: R^3 --> [0,1] and lets f(X_1, X_2, X_3) be her estimate of p. Okay, if that is what you meant. Bob does much the same with real random variables Y_i, i = 1, 2, 3 and function g: R^3 --> [0,1] and lets his estimate of p be g(Y_1, Y_2, Y_3). And Bob observes Y_1 = 0, Y_2 = 0, and Y_3 = 1. If functions f and g are the same, then, in this case, that is, with the data Alice and Bob observed, Alice and Bob get the same estimate for p. Else if f and g are different, then, even if the data they observe is the same, they might get different estimates. Even if f = g, since each of Alice and Bob is flipping the coin for themselves, they need not get the same estimate for p. Note: Since f is Borel measurable, we can set real random variable Z = f(X_1, X_2, X_3) and ask for E[Z], etc. Sometimes this step is useful. That is, our estimator of p is also a random variable. We might like to have E[Z] = p; in this case, Z is an 'unbiased' estimator of p. Also we can ask for the variance of Z, say, Var(Z), and maybe we want Var(Z) to be small. If we can show that Var(Z) is the smallest among all Borel measurable f, then the choice Alice made for f is a 'minimum variance' estimator. If Alice gets charged money for being wrong, then we can try to minimize the expected value of what Alice gets charged, and here we have a case of 'statistical decision theory'. Since order statistics are always sufficient, with the assumptions we have, Alice need only be told 2 zeros and 1 1 and can f'get about the rest. I see no surprises or difficulties here. But the sample space Omega is the same for both Alice and Bob. And, for both Alice and Bob, there is only one trial, that is, only one point little omega, in Omega involved. That is, in more detail, X_1 is a function, the function X_1: Omega --> R, and in our case for our trial little omega we have X_1(little omega) = 0. That is, usually in the notation we suppress little omega. Alice and Bob are both using the same trial little omega and the same sigma algebra script F on the same sample space Omega. We have no reason to believe that random variables X_1: Omega --> R and Y_1: Omega --> R are equal. Thus, given a Borel subset K of R, the events X_1^{-1}(K) in the sigma algebra script F on Omega and Y_1^{-1}(K) also in the sigma algebra script F on Omega need not be the same. That's a little of how applied probability based on 'modern probability' works. I see no problems and no need to consider frequentism or Bayesianism. Questions?

by graycat 12y ago

kgwgk 12y ago

Leaving aside that Kolmogorov's measure-theoretic approach is not the only axiomatic definition of probability (Cox's axioms yield a quite similar foundation, though with finite additivity only), you won't find anyone here that says there is a problem with the mathematical construction. The frequentist/Bayesian debate is related to the INTERPRETATION of probability. Kolmogorov won't help you to map the real world to the probability space. Let's say we have a loaded coin, we want to estimate the probability of getting tails (assume this is a i.i.d. random variable). Alice decides to keep throwing until she gets a tail: she gets the sequence HHT Bob decides to throw the coin three times: he gets the sequence HHT Alice takes her event A, her sigma-algebra, the whole shebang, and produces an interval estimate for p. Bob takes his event A (which happened to be the same) and his probability space (which is different because the experimental design is different), and produces a different estimate for p. If you think that getting different results from the same data makes sense, you might be a frequentist. If you think that it doesn't, you might be a Bayesian. If you think that the question is not relevant because you can't derive the answer from your axioms, you might at least understand what we're talking about.

graycat 12y ago
This is all mixed up. Alice has real random variables X_1 (borrowing TeX notation for a subscript), X_2, X_3. We assume that {X_i|i = 1, 2, 3} is independent. We assume that for some number p in [0,1], P(X_i = 0) = p -- Alice uses 0 for H and 1 for T. Alice observes that X_1 = 0, X_2 = 0, and X_3 = 1. Now, for the set of real numbers, R, Alice has a Borel measurable function f: R^3 --> [0,1] and lets f(X_1, X_2, X_3) be her estimate of p. Okay, if that is what you meant. Bob does much the same with real random variables Y_i, i = 1, 2, 3 and function g: R^3 --> [0,1] and lets his estimate of p be g(Y_1, Y_2, Y_3). And Bob observes Y_1 = 0, Y_2 = 0, and Y_3 = 1. If functions f and g are the same, then, in this case, that is, with the data Alice and Bob observed, Alice and Bob get the same estimate for p. Else if f and g are different, then, even if the data they observe is the same, they might get different estimates. Even if f = g, since each of Alice and Bob is flipping the coin for themselves, they need not get the same estimate for p. Note: Since f is Borel measurable, we can set real random variable Z = f(X_1, X_2, X_3) and ask for E[Z], etc. Sometimes this step is useful. That is, our estimator of p is also a random variable. We might like to have E[Z] = p; in this case, Z is an 'unbiased' estimator of p. Also we can ask for the variance of Z, say, Var(Z), and maybe we want Var(Z) to be small. If we can show that Var(Z) is the smallest among all Borel measurable f, then the choice Alice made for f is a 'minimum variance' estimator. If Alice gets charged money for being wrong, then we can try to minimize the expected value of what Alice gets charged, and here we have a case of 'statistical decision theory'. Since order statistics are always sufficient, with the assumptions we have, Alice need only be told 2 zeros and 1 1 and can f'get about the rest. I see no surprises or difficulties here. But the sample space Omega is the same for both Alice and Bob. And, for both Alice and Bob, there is only one trial, that is, only one point little omega, in Omega involved. That is, in more detail, X_1 is a function, the function X_1: Omega --> R, and in our case for our trial little omega we have X_1(little omega) = 0. That is, usually in the notation we suppress little omega. Alice and Bob are both using the same trial little omega and the same sigma algebra script F on the same sample space Omega. We have no reason to believe that random variables X_1: Omega --> R and Y_1: Omega --> R are equal. Thus, given a Borel subset K of R, the events X_1^{-1}(K) in the sigma algebra script F on Omega and Y_1^{-1}(K) also in the sigma algebra script F on Omega need not be the same. That's a little of how applied probability based on 'modern probability' works. I see no problems and no need to consider frequentism or Bayesianism. Questions?