How Random is a Coin Toss? Bayesian Inference and the Symbolic Dynamics of Deterministic Chaos

TL;DR

This paper introduces a Bayesian inference method for identifying generating partitions and estimating entropy rates in chaotic dynamical systems, enhancing understanding of unpredictability in nonlinear systems.

Contribution

It develops a Bayesian framework for analyzing symbolic dynamics, enabling the detection of Markov properties and entropy estimation from finite data samples.

Findings

Bayesian inference effectively identifies generating partitions.

Entropy rates can be accurately estimated from discretized data.

Method improves analysis of unpredictability in chaotic systems.

Abstract

Symbolic dynamics has proven to be an invaluable tool in analyzing the mechanisms that lead to unpredictability and random behavior in nonlinear dynamical systems. Surprisingly, a discrete partition of continuous state space can produce a coarse-grained description of the behavior that accurately describes the invariant properties of an underlying chaotic attractor. In particular, measures of the rate of information production--the topological and metric entropy rates--can be estimated from the outputs of Markov or generating partitions. Here we develop Bayesian inference for k-th order Markov chains as a method to finding generating partitions and estimating entropy rates from finite samples of discretized data produced by coarse-grained dynamical systems.