Bayesian Structural Inference for Hidden Processes

TL;DR

This paper presents a Bayesian method for inferring the structure of complex hidden processes using a set of candidate unifilar HMM topologies, enabling accurate estimation of process properties and uncertainty quantification.

Contribution

It introduces Bayesian Structural Inference (BSI) that guarantees models are epsilon-machines and provides analytic expressions for model inference from data.

Findings

BSI accurately estimates entropy rate and statistical complexity.

Posterior distribution over models better captures uncertainty.

Effective on finite, infinite, and out-of-class processes.

Abstract

We introduce a Bayesian approach to discovering patterns in structurally complex processes. The proposed method of Bayesian Structural Inference (BSI) relies on a set of candidate unifilar HMM (uHMM) topologies for inference of process structure from a data series. We employ a recently developed exact enumeration of topological epsilon-machines. (A sequel then removes the topological restriction.) This subset of the uHMM topologies has the added benefit that inferred models are guaranteed to be epsilon-machines, irrespective of estimated transition probabilities. Properties of epsilon-machines and uHMMs allow for the derivation of analytic expressions for estimating transition probabilities, inferring start states, and comparing the posterior probability of candidate model topologies, despite process internal structure being only indirectly present in data. We demonstrate BSI's effectiveness in estimating a process's randomness, as reflected by the Shannon entropy rate, and its structure, as quantified by the statistical complexity. We also compare using the posterior distribution over candidate models and the single, maximum a posteriori model for point estimation and show that the former more accurately reflects uncertainty in estimated values. We apply BSI to in-class examples of finite- and infinite-order Markov processes, as well to an out-of-class, infinite-state hidden process.