Barkley Rosser, what I have in mind is a reality which in principle predictable given enough information. So there is a “true” distribution—it’s conditional on information which specifies the state of the world exactly, so it’s a delta function at whatever the observables actually turn out to be. Now, there exists unbounded sequences of bits which don’t settle down to any particular relative frequency over the long run, and likewise, there is no guarantee that any particular sequence of observed data will lead to my posterior distribution getting closer and closer to one particular point in parameter space—if my model doesn’t at least partially account for the information which determines what values the observables take. Then I wave my hands and say, “That doesn’t seem to happen a lot in practical applications, or at least, when it does happen we humans don’t publish until we’ve improved the model to the point of usefulness.”
I didn’t follow your point about a distribution for which Bayes’ Theorem doesn’t hold. Are you describing a joint probability distribution for which Bayes’ Theorem doesn’t hold, or are you talking about a Bayesian modeling problem in which Bayes estimators are inconsistent a la Diaconis and Freedman, or do you mean something else again?
Barkley Rosser, what I have in mind is a reality which in principle predictable given enough information. So there is a “true” distribution—it’s conditional on information which specifies the state of the world exactly, so it’s a delta function at whatever the observables actually turn out to be. Now, there exists unbounded sequences of bits which don’t settle down to any particular relative frequency over the long run, and likewise, there is no guarantee that any particular sequence of observed data will lead to my posterior distribution getting closer and closer to one particular point in parameter space—if my model doesn’t at least partially account for the information which determines what values the observables take. Then I wave my hands and say, “That doesn’t seem to happen a lot in practical applications, or at least, when it does happen we humans don’t publish until we’ve improved the model to the point of usefulness.”
I didn’t follow your point about a distribution for which Bayes’ Theorem doesn’t hold. Are you describing a joint probability distribution for which Bayes’ Theorem doesn’t hold, or are you talking about a Bayesian modeling problem in which Bayes estimators are inconsistent a la Diaconis and Freedman, or do you mean something else again?