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Mailing List complex-science@necsi.org Message #6439 | |
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Aleks --I hope you understand that the body of work on this subject is huge, and I You presented what I believe was an inaccurate view of the differences between the two main approaches to Probability. I responded with what I thought was a more accurate description of the differences. The two books I cited are not books which describe my own point of view, but books which present these differences neutrally -- in the first case (Gillies), with respect to the underlying philosophies, in the second case (Klir and Wierman), with respect to mathematical formalisms. Of course, if you reject any alternative to Bayesianism, as Jaynes does, then you would consider any book which even describes these other approaches as biased.
Because this is the underlying difference between Bayesian approaches and frequentist approaches. which makes it seem that Bayesian statistics is all subjectivity, Or it can be considered a totally inappropriate aspect to statistical inference, as most frequentists believe. The point here is that these 2 approaches are different paradigms. What one focuses on, the other thinks is invalid. Frequentists do not use them, but the consequence is equivalent Not at all. It is only "equivalent" inside a Bayesian framework. If you reject this framework, it is not equivalent.
And, likewise, there are problems to which the Bayesian approach fails you. I refer you to the Gillies book, and to the books of Deborah Mayo. (Her books *are* strong propoponents for the frequentist viewpoint, unlike Gillies.)
Because the key difference, IMO, between the two approaches is not the assumption of crisp or vague reality, but whether or not it is valid to consider probabilities of statements for which no objective evidence may be produced. I guess we disagree on this point.
Your post said: Here, I stand by my comment in response.
I did not "diss" Bayesian statistics on the basis of priors. I criticized it for accepting as valid, statements for which there can be no objective evidence produced. In any reasonable definition of science, we would expect experiments to be repeatable, and claims not to depend upon the internal, subjective beliefs of the person making the claim. In that sense, Bayesian approaches are not scientific, since they lack such objectivity. If I said to you that I believe that the probability of reincarnation was 95%, what experiment could you conduct to enable this prior probability to be revised? Why then would my statement be accepted as scientific? There are still problems if we consider a statement for which evidence could be produced: eg, I believe that the probability of life being found on Mars within the next 300 years is 30%. Why should my prior beliefs have anything to do with the act of inference? Basing a theory of inference on the subjective beliefs of the person doing the inferencing seems contrary to commonly-understood notions of science, IMHO. And, what happens when scientists have different prior beliefs? Where is the Bayesian theory for the coherent combination of different prior probabilities? The usual approach is to rely on theorems which show that different prior probabilities will converge to the same posterior probability in the case when all participants receive the same information and process it in the same way. But this rarely if ever happens, and, in any case, the mathematics (again!) relies on very restrictive assumptions. I conclude from this that Bayesianism has yet to demonstrate, beyond reasonable doubt, that it is a science. it probably doesn't matter that there are good Bayesian You misunderstand the criticism re LEM. Here's an example: You are policeman investigating a crime. You have some evidence that person A did the crime. You also have some other evidence that person A did not do the crime. The evidence you have points to both these statements ("A did the crime" and "A did not do the crime") being true simultaneously. This contradicts LEM. In order to decide which statement to believe, you need to compare the two sets of evidence. But what if the evidence itself has uncertain elements, over which one could place probabilities? The recent alternatives to Probability -- Possibility theory, Dempster-Shafer theory -- cope with this situation better than does Probability theory. It is possible to formalize these types of problem in second- or higher-order Probability theory (probabilities about probabilities). But this quickly becomes unintuitive, as I mentioned. Human decision makers understand Possibility and D-S approaches more readily than they do second-order probabilities.
Both Cox and di Finetti seek to justify the use of Probability theory by mathematical arguments. In both cases, they make assumptions which do not apply in all application domains. As I said, the point made by the critics of Probability is that it is a model of reality, and, as with any model, it captures some aspects of reality and misses others. Strong protagonists of Probability theory, such as Cox and Jaynes, seem not to understand this. I think this idea is fundamental in AI, which is perhaps why people in AI have led the search for alternatives to classical Probability.
I was not criticizing you. I was trying to explain why Lotfi Zadeh is held in low regard by many statisticians, and yet revered by many AI people and control engineers. -- Peter |
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