Bayesian criteria based on universal measures

In the minimum description length (MDL) and Bayesian criteria, we construct description length of data zⁿ = z₁⋯z_n of length n such that the length divided by n almost converges to its entropy rate as n → ∞, assuming z_i is in a finite set A. In model selection, if we knew the true probability P of zⁿ ϵ Aⁿ, we would choose a model F such that the posterior probability of F given zⁿ is maximized. But, in many situations, we use Q : Aⁿ → [0,1] such that Σ(zⁿ) _ϵ (Aⁿ) Q(zⁿ) ≤ 1 rather than P because only data zⁿ are available. In this paper, we consider an extension such that each of the attributes in data can be either discrete or continuous. The main issue is what Q is qualified to be an alternative to P in the generalized situations. We propose the condition in terms of the Radon-Nikodym derivative of P with respect to Q, and give the procedure of constructing Q in the general setting. As a result, we obtain the MDL/Bayesian criteria in a general sense.