Convergence and Loss Bounds for Bayesian Sequence Prediction

Abstract

The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with Bayes' rule if the true generating distribution $μ$ of the sequences $x_1x_2x_3...$ is known. If $μ$ is unknown, but known to belong to a class $M$ one can base ones prediction on the Bayes mix $ξ$ defined as a weighted sum of distributions $ν\in M$. Various convergence results of the mixture posterior $ξ_t$ to the true posterior $μ_t$ are presented. In particular a new (elementary) derivation of the convergence $ξ_t/μ_t\to 1$ is provided, which additionally gives the rate of convergence. A general sequence predictor is allowed to choose an action $y_t$ based on $x_1...x_{t-1}$ and receives loss $\ell_{x_t y_t}$ if $x_t$ is the next symbol of the sequence. No assumptions are made on the structure of $\ell$ (apart from being bounded) and $M$. The Bayes-optimal prediction scheme $Λ_ξ$ based on mixture $ξ$ and the Bayes-optimal informed prediction scheme $Λ_μ$ are defined and the total loss $L_ξ$ of $Λ_ξ$ is bounded in terms of the total loss $L_μ$ of $Λ_μ$. It is shown that $L_ξ$ is bounded for bounded $L_μ$ and $L_ξ/L_μ\to 1$ for $L_μ\to \infty$. Convergence of the instantaneous losses are also proven.