Similarity-Driven Proposals for MCMC Algorithms on Discrete Spaces

2026-05-20

AI Summary

This work addresses the challenge of inefficient posterior exploration in hierarchical discrete models with latent variables, where conventional MCMC methods struggle due to the need to integrate out latent variables. The authors propose a similarity-driven MCMC approach that constructs a proposal mechanism based on a data-driven measure of discrepancy between observations and model predictions, thereby guiding transitions toward regions of higher posterior support without explicitly integrating latent variables. This method represents the first application of similarity-driven proposals to discrete-space MCMC and is naturally suited to complex hierarchical discrete models. Experiments on both synthetic and real-world data demonstrate substantial improvements in sampling efficiency and posterior exploration, confirming its effectiveness in models such as Dirichlet–Multinomial regression.

Abstract

Recent research has led to the development of MCMC algorithms with likelihood-informed proposals when targeting posterior distributions supported on discrete state spaces. Our work is placed within this field and puts forward a new MCMC methodology based upon similarity-driven proposals. Such proposals sway transitions towards states favored by the posterior via use of a data-driven measure of discrepancy between observations and the proposed model. Our approach can naturally cover classes of hierarchical models that involve both discrete variables and additional latent ones, without a requirement of integrating our the latter, in contrast to previous works in this field. The new algorithms are illustrated in simulation settings and in a involved real data scenario with a Dirichlet-Multinomial regression model.

Problem

• MCMC
• discrete spaces
• posterior sampling
• hierarchical models
• latent variables

Innovation

• similarity-driven proposals
• MCMC
• discrete spaces
• likelihood-informed
• hierarchical models