This post is an attempt to gather people to solve a particular problem in mathematics, something that can actually be published and seems to me simple enough to test this mathoverflow as a problem-solving environment, in the spirit of the Polymath project.
Bayesian probabilities are usually justified by the Cox theorems, that can be written this way:
Under some technical assumptions (continuity, etc, etc...), given a set $P$ of objects $A, B, C, \ldots$, with a boolean algebra defined over it with operations $A \vee B$ (and) and $A | B$ (or) such that:
1) $A \vee B = B \vee A$
2) $A \vee (B \vee C) = (A \vee B) \vee C$
3) $A | (B \vee C) = (A|B) \vee (A|C)$
and a "valuation":
$f : P \rightarrow \mathcal{R}$
there is a strictly monotonic "regraduation function" $R : \mathcal{R} \rightarrow \mathcal{R}$ such that:
$R(f(A\vee R(f(A | B)) = R(f(A)) + R(f(B))$ (sum rule)
and
$R(f(A|B)) R(f(A\vee B)) = R(f(A) ) R(f(B))$ (product rule)
This theorem allows one to show that any system designed to "evaluate" boolean expressions consistently with a single real number redunds in the laws of classical probability (this can be seen shortly here: arxiv:physics/0403089 and more thoroughly here: arxiv:abs/0808.0012)
Recently this has been extended for valuations of the type $f : P \rightarrow \mathcal{R}^2$ in http://arxiv.org/abs/0907.0909 and they proved that there are just 5 canonical valuations compatible with the underlying Boolean algebra (one of them giving a complex field structure to the "valuation" field).
My question/proposal is: is it possible/interesting/feasible to classify at least a class of valuations of the type:
$f : P \rightarrow W$
where W is a continuous manifold (a "valuation field", like $\mathcal{R}$ in the original Cox theorems)? If we retrict our attention to $\mathcal{R}^n$ for example, is there, for each n, a set of canonical valuations to which all others can be reduced after a regraduation?
If this can be done, are those nice rules for inference in some sense? Are they useful as inference tools in specific situations?
How would you start to attack this problem?
