Followup to: Bayesian Flame.

This post is a chronicle of my attempts to understand Cyan's #2. (Bayesian Flame was an approximate parse of #1.) Warning: long, some math, lots of links, probably lots of errors. At the very least I want this to serve as a good reference for further reading.

Introduction

To the mathematical eye, many statistical problems share the following minimal structure:

A space of parameters. (Imagine a freeform blob without assuming any metric or measure.)
A space of possible outcomes. (Imagine another, similarly unstructured blob.)
Each point in the parameter space determines a probability measure on the outcome space.

By itself, this kind of input is too sparse to yield solutions to statistical problems. What additional structure on the spaces should we introduce?

The answer that we all know and love

Assuming some "prior" probability measure on the parameter space yields a solution that's unique, consistent and wonderful in all sorts of ways. This has led some people to adopt the "subjectivist" position saying priors are so basic that they ought not be questioned. One of its most prominent defenders was Leonard Jimmie Savage who put forward the following argument:

Suppose, for example, that the person is offered an even-money bet for five dollars - or, to be ultra-rigorous, for five utiles - that internal combustion engines in American automobiles will be obsolete by 1970. If there is any event to which an objectivist would refuse to attach probability, that corresponding to the obsolescence in question is one... Yet, I think I may say without presumption that you would regard the bet against obsolescence as a very sound investment.

This is a fine argument for using priors when you're betting money, but there's a snag: however much you are willing to bet, this doesn't give you grounds to publish papers about the future that you inferred from your intuitive prior! Any apriori information used in science should be justified for scientific objectivity.

(At this point Eliezer raises the suggestion that scientists ought to communicate with likelihood ratios only. That might be a brave new world to live in; too bad we'll have to stop teaching kids that g approximately equals 9.8 m/s² and give them likelihood profiles instead.)

Rather than dive deeper into the fascinating topic of "uninformative priors", let's go bac...