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Consider universal data compression: the length $l(x^n)$ of sequence $x^n\inA^n$ with finite alphabet $A$ and length $n$ satisfies Kraft's inequality over$A^n$, and $-\frac{1}{n}\log \frac{P^n(x^n)}{Q^n(x^n)}$ almost surely convergesto zero as $n$ grows for the $Q^n(x^n)=2^{-l(x^n)}$ and any stationary ergodicsource $P$. In this paper, we say such a $Q$ is a universal Bayesian measure.We generalize the notion to the sources in which the random variables may beeither discrete, continuous, or none of them. The basic idea is due to BorisRyabko who utilized model weighting over histograms that approximate $P$,assuming that a density function of $P$ exists. However, the range of $P$depends on the choice of the histogram sequence. The universal Bayesian measureconstructed in this paper overcomes the drawbacks and has many applications toinfer relation among random variables, and extends the application area of theminimum description length principle.

Universal Bayesian Measures and Universal Histogram Sequences