tag:blogger.com,1999:blog-8666091.post113595314244958426..comments2014-12-18T22:47:26.740+01:00Comments on The Reference Frame: Bayesian probability ILuboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-8666091.post-1135977387557824412005-12-30T22:16:00.000+01:002005-12-30T22:16:00.000+01:00You said:It is much more difficult when you are ma...You said:<BR/><BR/><I>It is much more difficult when you are making bets about some events whose exact probabilities are unknown. Even in these cases, we often like to say a number that expresses our beliefs quantitatively. Such a notion of probability is called Bayesian probability and it does not really belong to exact sciences.</I><BR/><BR/>I have seen this style of argument against Bayes before. It wrongly assumes that Bayes is about <I>defining</I> probabilities rather than <I>manipulating</I> them.<BR/><BR/>The Bayesian approach is about doing inference by manipulating joint probabilities in specific ways (e.g. Bayes theorem). Bayes doesn't really care where its prior probabilities come from, and the approach does <I>not</I> actually define a method for supplying these priors to bootstrap the inference process.<BR/><BR/>In contrast to your claim above, Bayes is perfectly happy to use experimentally measured <I>frequentist</I> probabilities as priors to bootstrap its inference process, but there are <I>more</I> rigorous ways of using such frequentist data in the Bayesian approach.<BR/><BR/>The following paper gives a nice axiomatic approach to Bayesian inference, which you might find interesting:<BR/><BR/>Cox R T, Probability, frequency and reasonable expectation, Am. J. Phys., 1946, 14(1), 1-13.<BR/><BR/>Essentially, the paper shows that the Bayesian way of manipulating joint probabilities is the only consistent way of doing inference. I have not got a copy of the paper to hand, but I seem to recall that it doesn't mention the word Bayes anywhere in it!Stevehttp://www.blogger.com/profile/16386879829062354101noreply@blogger.com