Geometry of Higher-Order Markov Chains

TL;DR

This paper characterizes the algebraic structure of higher-order Markov chain models, providing explicit Gr"obner bases and geometric descriptions, revealing their underlying varieties and likelihood landscape.

Contribution

It offers an explicit algebraic description of the prime ideal for higher-order Markov chain models, including binary and non-binary cases, using Gr"obner bases and geometric analysis.

Findings

For binary states, the model is a linear space with two local maxima.

In the non-binary case, the model is a cone over a Segre variety.

Explicit Gr"obner basis for the prime ideal is provided.

Abstract

We determine an explicit Gr\"obner basis, consisting of linear forms and determinantal quadrics, for the prime ideal of Raftery's mixture transition distribution model for Markov chains. When the states are binary, the corresponding projective variety is a linear space, the model itself consists of two simplices in a cross-polytope, and the likelihood function typically has two local maxima. In the general non-binary case, the model corresponds to a cone over a Segre variety.