Prediction and Generation of Binary Markov Processes: Can a Finite-State Fox Catch a Markov Mouse?

TL;DR

This paper derives the minimal generative models for binary Markov processes, revealing that different parameter regions require distinct topologies and that prediction and generation costs vary, especially for nearly independent processes.

Contribution

It introduces the minimal generator for arbitrary binary Markov processes, clarifies the topologies needed, and compares the costs of prediction versus generation.

Findings

Three distinct topologies are needed for different parameter regions.

A previously proposed generator is not minimal.

The difference between prediction and generation costs peaks near independent processes.

Abstract

Understanding the generative mechanism of a natural system is a vital component of the scientific method. Here, we investigate one of the fundamental steps toward this goal by presenting the minimal generator of an arbitrary binary Markov process. This is a class of processes whose predictive model is well known. Surprisingly, the generative model requires three distinct topologies for different regions of parameter space. We show that a previously proposed generator for a particular set of binary Markov processes is, in fact, not minimal. Our results shed the first quantitative light on the relative (minimal) costs of prediction and generation. We find, for instance, that the difference between prediction and generation is maximized when the process is approximately independently, identically distributed.