The Past and the Future in the Present

TL;DR

This paper explores how the excess entropy, representing shared information between past and future, originates from present causal states, and discusses challenges in its estimation.

Contribution

It provides a detailed proof linking excess entropy to causal states and highlights estimation challenges in information analysis.

Findings

Excess entropy derives from present causal states.

Estimation of excess entropy involves subtle problems.

The paper clarifies the relationship between past, present, and future information.

Abstract

We show how the shared information between the past and future---the excess entropy---derives from the components of directional information stored in the present---the predictive and retrodictive causal states. A detailed proof allows us to highlight a number of the subtle problems in estimation and analysis that impede accurate calculation of the excess entropy.