Prediction, Retrodiction, and The Amount of Information Stored in the Present

TL;DR

This paper presents a time-symmetric framework for stochastic dynamical systems, linking prediction and retrodiction to measure system organization and introduce new invariants like crypticity and causal irreversibility.

Contribution

It introduces a unified, time-symmetric representation of stochastic systems that connects past, future, and present information measures, revealing new invariants and computational methods.

Findings

Excess entropy equals mutual information between predictive and retrodictive states.

New invariants: crypticity and causal irreversibility.

Unified time-symmetric representation of system information.

Abstract

We introduce an ambidextrous view of stochastic dynamical systems, comparing their forward-time and reverse-time representations and then integrating them into a single time-symmetric representation. The perspective is useful theoretically, computationally, and conceptually. Mathematically, we prove that the excess entropy--a familiar measure of organization in complex systems--is the mutual information not only between the past and future, but also between the predictive and retrodictive causal states. Practically, we exploit the connection between prediction and retrodiction to directly calculate the excess entropy. Conceptually, these lead one to discover new system invariants for stochastic dynamical systems: crypticity (information accessibility) and causal irreversibility. Ultimately, we introduce a time-symmetric representation that unifies all these quantities, compressing the two directional representations into one. The resulting compression offers a new conception of the amount of information stored in the present.