The convergence rate of the Gibbs sampler for generalized $1-$D Ising model

TL;DR

This paper analyzes the convergence rate of the Gibbs sampler for the generalized 1D Ising model, providing improved bounds on the second largest eigenvalue of its transition matrix.

Contribution

It generalizes previous bounds for the 1D Ising model's convergence rate using Diaconis and Stroock's method, improving upon Ingrassia's earlier results.

Findings

Derived a new bound for the second largest eigenvalue of the transition matrix.

Extended previous results from two-state models to generalized models.

Improved the theoretical understanding of Gibbs sampler convergence in 1D Ising models.

Abstract

The rate of convergence of the Gibbs sampler for the generalized one-dimensional Ising model is determined by the second largest eigenvalue of its transition matrix in absolute value denoted by $\beta^*$. In this paper we generalize a result from Shiu and Chen $(2015)$ for the one-dimensional Ising model with two states which gives a bound for $\beta^*$ . The method is based on Diaconis and Stroock bound for reversible Markov processes. The new bound presented in this paper improves Ingrassia's $(1994)$ result.