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Nothing about this reply makes me reconsider labeling it "naive frequentism." I think you'll find that it's hard to do even bread-and-butter statistical tasks l

Nothing about this reply makes me reconsider labeling it "naive frequentism." I think you'll find that it's hard to do even bread-and-butter statistical tasks like deciding how many subjects to include in a randomized trial without moving beyond axiomatic probability.

graycat 12y ago
Sorry, it appears that we just are a long way from communicating clearly. I'll try again, a little: What I outlined are the axioms of probability as started by Kolmogorov in 1933 and based on Lebesgue's measure theory (of near 1900). So, Kolmogorov finds something in Lebesgue's theory, that is, a particular measure space that has everything we want for probability theory. In this way Kolmogorov shows that we can regard probability theory as just another measure space in measure theory. Why do that? To get a different probability theory? Not really: Kolmogorov's start just gives essentially the same probability theory we had in 1932. It is just that in 1932, we had to talk about trials, events, probabilities, and random variables without being able to say, mathematically, what the heck they were. To see the importance here, back to near 1900 with B. Russell, etc. there was an effort to redefine everything in mathematics starting with just sets. Then everything was constructed starting with just the low level ideas of sets. The serious result was axiomatic set theory, e.g., as in P. Suppes, Axiomatic Set Theory. So, numbers, functions, calculus, lines, planes, spheres, groups, rings, fields, vector spaces. etc. were all defined based on sets. And similarly for Lebesgue's measure theory. Then, after Kolmogorov, probability theory was also defined based just on sets. Whew! For what you want to do with probability theory in statistics, etc. as you mentioned, Kolmogorov's axioms should be of zero concern to you except you might feel a little better knowing that the probability theory you have been using all along does have a solid foundation just on sets. So, you can go right along with applied probability as you have been doing, and Kolmogorov's axioms will essentially never get involved, never help you and never hurt you. For more, via the axioms, we can define trials, events, probabilities, random variables, conditional probability, stochastic processes, Markov processes, Gaussian random variables, Chi squared random variables, sufficient statistics, distributions, random vectors, and on and on that you have already been knowing, loving, and using. But, maybe I spoke too soon: For Markov processes, martingales, sufficient statistics, we very much want the Radon-Nikodym theorem of measure theory and use it for random variables, and we do. So, without the Kolmogorov's axioms, we would be somewhat stuck-o for Markov processes, etc. > I think you'll find that it's hard to do even bread-and-butter statistical tasks like deciding how many subjects to include in a randomized trial without moving beyond axiomatic probability. Not at all. There is no "moving beyond". Instead we proceed essentially as we might have in 1932. That is, in such work we rarely or never think about measurable spaces, measure spaces, sigma algebras, measurability, random variables as functions on the set of trials, etc. Again, we just continue to do applied probability and statistics essentially as we might have in 1932. Or, the Komogorov axioms are down in the sub, sub basement, and we rarely go there. For the rest of the structure, it is just the same or nearly so. Okay?
- grayclhn 12y ago
  I've studied probability theory, maybe more than you. I like it. But nothing from probability theory suggests why "size" and "power" might be useful properties in test statistics. Without other ideas like those, doing useful statistics would be pretty tough. I notice you didn't propose a way to choose the sample size for a randomized trial in your reply. I'd love to see it -- using only probability theory and nothing from the stats literature. ;)
  - 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.

graycat 12y ago

Sorry, it appears that we just are a long way from communicating clearly. I'll try again, a little: What I outlined are the axioms of probability as started by Kolmogorov in 1933 and based on Lebesgue's measure theory (of near 1900). So, Kolmogorov finds something in Lebesgue's theory, that is, a particular measure space that has everything we want for probability theory. In this way Kolmogorov shows that we can regard probability theory as just another measure space in measure theory. Why do that? To get a different probability theory? Not really: Kolmogorov's start just gives essentially the same probability theory we had in 1932. It is just that in 1932, we had to talk about trials, events, probabilities, and random variables without being able to say, mathematically, what the heck they were. To see the importance here, back to near 1900 with B. Russell, etc. there was an effort to redefine everything in mathematics starting with just sets. Then everything was constructed starting with just the low level ideas of sets. The serious result was axiomatic set theory, e.g., as in P. Suppes, Axiomatic Set Theory. So, numbers, functions, calculus, lines, planes, spheres, groups, rings, fields, vector spaces. etc. were all defined based on sets. And similarly for Lebesgue's measure theory. Then, after Kolmogorov, probability theory was also defined based just on sets. Whew! For what you want to do with probability theory in statistics, etc. as you mentioned, Kolmogorov's axioms should be of zero concern to you except you might feel a little better knowing that the probability theory you have been using all along does have a solid foundation just on sets. So, you can go right along with applied probability as you have been doing, and Kolmogorov's axioms will essentially never get involved, never help you and never hurt you. For more, via the axioms, we can define trials, events, probabilities, random variables, conditional probability, stochastic processes, Markov processes, Gaussian random variables, Chi squared random variables, sufficient statistics, distributions, random vectors, and on and on that you have already been knowing, loving, and using. But, maybe I spoke too soon: For Markov processes, martingales, sufficient statistics, we very much want the Radon-Nikodym theorem of measure theory and use it for random variables, and we do. So, without the Kolmogorov's axioms, we would be somewhat stuck-o for Markov processes, etc. > I think you'll find that it's hard to do even bread-and-butter statistical tasks like deciding how many subjects to include in a randomized trial without moving beyond axiomatic probability. Not at all. There is no "moving beyond". Instead we proceed essentially as we might have in 1932. That is, in such work we rarely or never think about measurable spaces, measure spaces, sigma algebras, measurability, random variables as functions on the set of trials, etc. Again, we just continue to do applied probability and statistics essentially as we might have in 1932. Or, the Komogorov axioms are down in the sub, sub basement, and we rarely go there. For the rest of the structure, it is just the same or nearly so. Okay?

grayclhn 12y ago
I've studied probability theory, maybe more than you. I like it. But nothing from probability theory suggests why "size" and "power" might be useful properties in test statistics. Without other ideas like those, doing useful statistics would be pretty tough. I notice you didn't propose a way to choose the sample size for a randomized trial in your reply. I'd love to see it -- using only probability theory and nothing from the stats literature. ;)
- 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.