# Reversible Jump Markov Chain Monte Carlo Methods

## Introduction

This web page contains supplementary material for the following project:

• Ai Jialin, Reversible Jump Markov Chain Monte Carlo Methods.
MSc thesis, University of Leeds, Department of Statistics, 2011/12.

## Abstract

This dissertation is about the reversible-jump Markov chain Monte Carlo (RJMCMC) method in Bayesian statistics and its application to solving the change-points problem. A change-points problem is to determine the parameters of a step model with ﬂexible number of steps. This model is widely used in different ﬁelds. Thus many application, such as change-points in genes determining and ﬂexible number’s variables selection in regression, beneﬁt from RJMCMC.

Firstly I introduce Markov chain Monte Carlo (MCMC) and the Metropolis-Hastings (MH) Algorithm. I explain the concept “reversibility”, give the relationship between “reversible” and “stationary”, and then prove that the Markov chain generated by M-H is reversible and stationary.

Secondly, I introduce Bayesian estimation and realise it with M-H algorithm. I gave a simple and clear example and implement it in R with Bayesian estimates algorithm. Next, I introduced the subspaces assumption of reversible-jump Markov chain, and the basic form of the acceptance probability jumping between subspaces of differing dimensionality based on Green (1995).

Finally, I try to solve a change-points problem of coal-mining disasters data. I calculate the acceptance probabilities of different move types, combine the results and the Metropolis Hastings algorithm into an R program. And I also analyse the reversible-jump Markov chain generated by the program.