I've been trying to understand Cox's Theorem and the problems surrounding it. There's so much information on this topic that I've become confused as to the exact state of the theorem. I've gathered there are three main issues, but since they cover a broad variety of topics I've split it up into multiple questions. I hope this is sufficiently narrowed down (the original question).

My main reference is K. Van Horn, A Guide to Cox's Theorem, 2003.

The assumption that a plausibility $(A|B)$ (A given B) can always be represented by a real number is often a point of controversy for various reasons (see here for the other two issues with this assumption).

The objection is raised that one-dimensional representations of plausibility can't adequately represent ignorance. Jaynes seems to deal with this using maximum entropy methods (Information Theory and Statistical Mechanics I & II, 1957) and prior transformation groups (Prior Transformations, 1968). From what I gathered this solution is not considered sufficient by many. Are there any specific objections?

Next, Van Horn mentions:

Another motivation for two-dimensional theories has been the perception that proper application of Bayesian methods requires knowledge of the ‘‘true’’ probabilities, which are viewed as physical properties. This can lead to theories in which one represents uncertainty as a convex set of probability distributions [20]. Jaynes dismisses such concerns about ‘‘true,’’ physical probabilities as examples of the ‘‘Mind Projection Fallacy’’ [21], yet even the Jaynesian viewpoint admits certain states of information that are mathematically equivalent to being uncertain about some ‘‘physical’’ probability. Bayesians deal with uncertain ‘‘physical’’ probabilities by reasoning about the probabilities of various physical probability values. This brings us back to the previous concern, representation of ignorance, and hence this second motivation for considering a two-dimensional theory reduces to the first.

[20] : H.E. Kyburg, Bayesian and Non-Bayesian Evidential Updating, 1985

[21] : E.T. Jaynes, Probability Theory as Logic, 1989

I'm not entirely sure why knowledge of ``true'' probabilities is necessary for proper application of Bayesian methods. Is this just the matter of devising a proper prior?

In summary:

  1. What are the problems with Jaynes' argument for objective priors and representing ignorance?

  2. What is the trouble concerning Jaynes' argument about the Mind Projection Fallacy?



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