I've made a number of updates to my website on Jaynes's book, _Probability Theory: The Logic of Science_ (http://leuther-analytics.com/jaynes/index.html). The most significant is an extended note on Jaynes's treatment of the Marginalization Paradox (Chapter 15), in which I conclude that Jaynes was wrong. As Jaynes does, I focused on the specific example of the change-point problem. Here are some of the things I show:
- - Bayesian B_2 can only obtain his result by the invalid step of taking a divergent integral; this divergent integral arises because the improper prior over \eta results in p(y | z) also being improper. - - If B_2 follows Jaynes's advice (which Jaynes unaccountably fails to follow in this discussion) of solving the problem with proper priors and then taking the limit of the solutions as the improper prior is approached, he gets the same answer as B_1. - - One way of seeing the source of the problem is that as we approach the improper prior over \eta, p(y | z) retains a significant probability mass in precisely the region where p(\zeta | y,z) is far from convergence. (This is an issue of non-uniform convergence.) If you have read Chapter 15 of PTLOS or are familiar with the Marginalization Paradox, I would greatly appreciate your feedback on what I have written. If you are reading PTLOS and have puzzled over the discussion of the MP, I hope what I have written will help clarify things.
