How Hidden are Hidden Processes? A Primer on Crypticity and Entropy Convergence

TL;DR

This paper explores the concept of crypticity in stationary processes, linking it to observer synchronization and entropy convergence, and introduces new geometric and classification tools to analyze hidden information in complex systems.

Contribution

It introduces a geometric framework for understanding crypticity, analyzes spin chains, and classifies processes based on cryptic and Markov orders, advancing the theoretical understanding of hidden processes.

Findings

Crypticity relates to observer synchronization.

Block-causal-state entropy is convex in block length.

Complete analysis of spin chains and classification scheme.

Abstract

We investigate a stationary process's crypticity---a measure of the difference between its hidden state information and its observed information---using the causal states of computational mechanics. Here, we motivate crypticity and cryptic order as physically meaningful quantities that monitor how hidden a hidden process is. This is done by recasting previous results on the convergence of block entropy and block-state entropy in a geometric setting, one that is more intuitive and that leads to a number of new results. For example, we connect crypticity to how an observer synchronizes to a process. We show that the block-causal-state entropy is a convex function of block length. We give a complete analysis of spin chains. We present a classification scheme that surveys stationary processes in terms of their possible cryptic and Markov orders. We illustrate related entropy convergence behaviors using a new form of foliated information diagram. Finally, along the way, we provide a variety of interpretations of crypticity and cryptic order to establish their naturalness and pervasiveness. Hopefully, these will inspire new applications in spatially extended and network dynamical systems.