# Estimation and selection for high-order Markov chains with Bayesian mixture transition distribution models

## TL;DR

This paper introduces Bayesian models for estimating and selecting high-order Markov chains using mixture transition distributions, enabling parsimonious modeling and order determination with shrinkage priors.

## Contribution

The paper proposes novel Bayesian models for high-order Markov chains that facilitate approximation, order selection, and interpretability through mixture transition distributions and shrinkage priors.

## Key findings

- Models effectively approximate high-order dynamics.
- Bayesian approach enables order and lag selection.
- Application demonstrates practical utility in ecological data.

## Abstract

We develop two models for Bayesian estimation and selection in high-order, discrete-state Markov chains. Both are based on the mixture transition distribution, which constructs a transition probability tensor with additive mixing of probabilities from first-order transition matrices. We demonstrate two uses for the proposed models: parsimonious approximation of high-order dynamics by mixing lower-order transition models, and order/lag selection through over-specification and shrinkage via priors for sparse probability vectors. The priors further shrink all models to an identifiable and interpretable parameterization, useful for data analysis. We discuss properties of the models and demonstrate their utility with simulation studies. We further apply the methodology to a data analysis from the high-order Markov chain literature and to a time series of pink salmon abundance in a creek in Alaska, U.S.A.