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Jump Markov Chains and Rejection-Free Metropolis Algorithms

J. S. Rosenthal, A. Dote, K. Dabiri, H. Tamura, S. Chen, A. Sheikholeslami

Abstract

We consider versions of the Metropolis algorithm which avoid the inefficiency of rejections. We first illustrate that a natural Uniform Selection Algorithm might not converge to the correct distribution. We then analyse the use of Markov jump chains which avoid successive repetitions of the same state. After exploring the properties of jump chains, we show how they can exploit parallelism in computer hardware to produce more efficient samples. We apply our results to the Metropolis algorithm, to Parallel Tempering, to a Bayesian model, to a two-dimensional ferromagnetic 4 x 4 Ising model, and to a pseudo-marginal MCMC algorithm.

Jump Markov Chains and Rejection-Free Metropolis Algorithms

Abstract

We consider versions of the Metropolis algorithm which avoid the inefficiency of rejections. We first illustrate that a natural Uniform Selection Algorithm might not converge to the correct distribution. We then analyse the use of Markov jump chains which avoid successive repetitions of the same state. After exploring the properties of jump chains, we show how they can exploit parallelism in computer hardware to produce more efficient samples. We apply our results to the Metropolis algorithm, to Parallel Tempering, to a Bayesian model, to a two-dimensional ferromagnetic 4 x 4 Ising model, and to a pseudo-marginal MCMC algorithm.

Paper Structure

This paper contains 13 sections, 13 theorems, 48 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Uniform Selection Algorithm
  3. The Jump Chain
  4. Using the Jump Chain for Estimation
  5. Application to the Metropolis Algorithm
  6. Alternating Chains
  7. Application to Parallel Tempering
  8. Numerical Examples
  9. A Bayesian Inference Problem with Real Data
  10. Simulations of an Ising Model
  11. A Pseudo-Marginal MCMC Example
  12. Summary
  13. Appendix: Proof of Proposition \ref{['appendixprop']}

Key Result

Proposition 1

If the Uniform Selection chain for Example 2 begins at state $4a$ for some positive integer $a \ge 2$, then the probability it will ever reach the state 3 is $\le (8/9)^{a-1} < 1$.

Figures (10)

  • Figure 1: The target distribution for Example 1.
  • Figure 2: The Metropolis chain for Example 1.
  • Figure 3: The Uniform Selection chain for Example 1.
  • Figure 4: Part of the target distribution for Example 2.
  • Figure 5: The Metropolis chain for Example 2.
  • ...and 5 more figures

Theorems & Definitions (24)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Remark 6
  • ...and 14 more