Bayesian criteria based on universal measures

In the minimum description length (MDL) and Bayesian criteria, we construct description length of data z
ⁿ
= z
₁
⋯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 z
ⁿ
ϵ A
ⁿ
, we would choose a model F such that the posterior probability of F given z
ⁿ
is maximized. But, in many situations, we use Q : A
ⁿ
→ [0,1] such that Σ(z
ⁿ
)
_ϵ
(A
ⁿ
) Q(z
ⁿ
) ≤ 1 rather than P because only data z
ⁿ
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.