Prediction from compression for models with infinite memory, with applications to hidden Markov and renewal processes

TL;DR

This paper develops optimal prediction methods for models with long-term memory, such as hidden Markov and renewal processes, achieving near-optimal risk bounds by extending universal compression ideas.

Contribution

It introduces a new estimator with risk bounds tied to redundancy and long-range dependencies, improving prediction accuracy for models with infinite memory.

Findings

Optimal prediction risk for HMMs and renewal processes determined.

Polynomial-time estimator for HMMs achieves b8(log n/n) risk.

Matching minimax lower bounds established via redundancy and mutual information.

Abstract

Consider the problem of predicting the next symbol given a sample path of length n, whose joint distribution belongs to a distribution class that may have long-term memory. The goal is to compete with the conditional predictor that knows the true model. For both hidden Markov models (HMMs) and renewal processes, we determine the optimal prediction risk in Kullback- Leibler divergence up to universal constant factors. Extending existing results in finite-order Markov models [HJW23] and drawing ideas from universal compression, the proposed estimator has a prediction risk bounded by redundancy of the distribution class and a memory term that accounts for the long-range dependency of the model. Notably, for HMMs with bounded state and observation spaces, a polynomial-time estimator based on dynamic programming is shown to achieve the optimal prediction risk {\Theta}(log n/n); prior to this work, the only known result of this type is O(1/log n) obtained using Markov approximation [Sha+18]. Matching minimax lower bounds are obtained by making connections to redundancy and mutual information via a reduction argument.