Limits on Inferring the Past

TL;DR

This paper explores the theoretical limits of inferring past states of dynamical systems, establishing relationships between retrodiction and thermodynamic entropy, and analyzing how system dynamics influence retrodictability.

Contribution

It introduces a formal framework for retrodictive inference, derives key equations linking entropy measures, and applies these to Langevin and logistic map systems to understand retrodictability.

Findings

Retrodiction entropy equals thermodynamic entropy at equilibrium.

Retrodictability behavior depends on the energy landscape, either decreasing or approaching a fixed value.

Maximal retrodictability occurs near bifurcations and during the transition to chaos.

Abstract

Here we define and study the properties of retrodictive inference. We derive equations relating retrodiction entropy and thermodynamic entropy, and as a special case, show that under equilibrium conditions, the two are identical. We demonstrate relations involving the KL-divergence and retrodiction probability, and bound the time rate of change of retrodiction entropy. As a specific case, we invert various Langevin processes, inferring the initial condition of \(N\) particles given their final positions at some later time. We evaluate the retrodiction entropy for Langevin dynamics exactly for special cases, and find that one's ability to infer the initial state of a system can exhibit two possible qualitative behaviors depending on the potential energy landscape, either decreasing indefinitely, or asymptotically approaching a fixed value. We also study how well we can retrodict points that evolve based on the logistic map. We find singular changes in the retrodictivity near bifurcations. Counterintuitively, the transition to chaos is accompanied by maximal retrodictability.