# Quantifying Uncertainty in Transdimensional Markov Chain Monte Carlo Using Discrete Markov Models

## TL;DR

This paper introduces a novel method to quantify the uncertainty in model probabilities estimated by transdimensional MCMC, using a Markov model approach to improve accuracy and assess effective sample size.

## Contribution

It proposes a new approach to estimate the precision of model probabilities in transdimensional MCMC using a transition matrix-based Markov model.

## Key findings

- Accurately assesses uncertainty in posterior model probabilities
- Provides estimates for effective sample size of MCMC output
- Demonstrates effectiveness in two model-selection examples

## Abstract

Bayesian analysis often concerns an evaluation of models with different dimensionality as is necessary in, for example, model selection or mixture models. To facilitate this evaluation, transdimensional Markov chain Monte Carlo (MCMC) relies on sampling a discrete indexing variable to estimate the posterior model probabilities. However, little attention has been paid to the precision of these estimates. If only few switches occur between the models in the transdimensional MCMC output, precision may be low and assessment based on the assumption of independent samples misleading. Here, we propose a new method to estimate the precision based on the observed transition matrix of the model-indexing variable. Assuming a first order Markov model, the method samples from the posterior of the stationary distribution. This allows assessment of the uncertainty in the estimated posterior model probabilities, model ranks, and Bayes factors. Moreover, the method provides an estimate for the effective sample size of the MCMC output. In two model-selection examples, we show that the proposed approach provides a good assessment of the uncertainty associated with the estimated posterior model probabilities.