Universal Bayesian Measures and Universal Histogram Sequences

Abstract

Consider universal data compression: the length $l(x^n)$ of sequence $x^n\in A^n$ 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 converges to zero as $n$ grows for the $Q^n(x^n)=2^{-l(x^n)}$ and any stationary ergodic source $P$. 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 be either discrete, continuous, or none of them. The basic idea is due to Boris Ryabko 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 measure constructed in this paper overcomes the drawbacks and has many applications to infer relation among random variables, and extends the application area of the minimum description length principle.