Learning the ergodic decomposition

TL;DR

This paper proves that a Bayesian agent observing a stationary process can asymptotically predict future outcomes as if it knew the process's long-term empirical frequencies, bridging Bayesian learning and ergodic theory.

Contribution

It establishes a theoretical link between Bayesian prediction and ergodic decomposition for stationary processes.

Findings

Bayesian predictions converge to those based on empirical frequencies

The results connect Bayesian inference with ergodic theory

Predictions become accurate over time for stationary processes

Abstract

A Bayesian agent learns about the structure of a stationary process from ob- serving past outcomes. We prove that his predictions about the near future become ap- proximately those he would have made if he knew the long run empirical frequencies of the process.