title: How hidden are hidden processes? A primer on crypticity and entropy convergence.
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. This is also a first step in developing applications in spatially extended and network dynamical systems.
authors.affiliation: Physics Department, University of California at Merced, 5200 North Lake Road, Merced, California 95343, USA. jmahoney3@ucmerced.edu
authors.forename: John R
authors.initials: JR
authors.lastname: Mahoney
authors.forename: Christopher J
authors.initials: CJ
authors.lastname: Ellison
authors.forename: Ryan G
authors.initials: RG
authors.lastname: James
authors.forename: James P
authors.initials: JP
authors.lastname: Crutchfield