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Efficiency of reversible MCMC methods: elementary derivations and applications to composite methods

Radford M. Neal, Jeffrey S. Rosenthal

TL;DR

The paper addresses comparing the efficiency of reversible Markov chain Monte Carlo on finite state spaces by the asymptotic variance $v(f,P)$. It develops an elegant set of equivalent conditions linking efficiency-dominance to eigenvalues and to operator expressions such as $P(I-P)^{-1}$, with $v(f,P)$ expressible as $v(f,P)=\sum_{i\ge2} (a_i)^2 \frac{1+\lambda_i}{1-\lambda_i}$. It then extends these ideas to composite schemes, showing that improvements to components transfer to the whole chain under suitable conditions, and discusses antithetic sampling as a case where efficiency can exceed i.i.d. sampling. The work provides self-contained, elementary proofs (including an accessible proof of Lemma ['effequivlemma']) and re-derives Peskun's theorem, offering practical guidelines for designing efficient MCMC via the combination of reversible updates and blockwise improvements.

Abstract

We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. We say that a chain on a finite state space with transition matrix $P$ efficiency-dominates one with transition matrix $Q$ if for every function of state it has lower (or equal) asymptotic variance. We give elementary proofs of some previous results regarding efficiency dominance, leading to a self-contained demonstration that a reversible chain with transition matrix $P$ efficiency-dominates a reversible chain with transition matrix $Q$ if and only if none of the eigenvalues of $Q-P$ are negative. This allows us to conclude that modifying a reversible MCMC method to improve its efficiency will also improve the efficiency of a method that randomly chooses either this or some other reversible method, and to conclude that improving the efficiency of a reversible update for one component of state (as in Gibbs sampling) will improve the overall efficiency of a reversible method that combines this and other updates. It also explains how antithetic MCMC can be more efficient than i.i.d. sampling. We also establish conditions that can guarantee that a method is not efficiency-dominated by any other method.

Efficiency of reversible MCMC methods: elementary derivations and applications to composite methods

TL;DR

The paper addresses comparing the efficiency of reversible Markov chain Monte Carlo on finite state spaces by the asymptotic variance . It develops an elegant set of equivalent conditions linking efficiency-dominance to eigenvalues and to operator expressions such as , with expressible as . It then extends these ideas to composite schemes, showing that improvements to components transfer to the whole chain under suitable conditions, and discusses antithetic sampling as a case where efficiency can exceed i.i.d. sampling. The work provides self-contained, elementary proofs (including an accessible proof of Lemma ['effequivlemma']) and re-derives Peskun's theorem, offering practical guidelines for designing efficient MCMC via the combination of reversible updates and blockwise improvements.

Abstract

We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. We say that a chain on a finite state space with transition matrix efficiency-dominates one with transition matrix if for every function of state it has lower (or equal) asymptotic variance. We give elementary proofs of some previous results regarding efficiency dominance, leading to a self-contained demonstration that a reversible chain with transition matrix efficiency-dominates a reversible chain with transition matrix if and only if none of the eigenvalues of are negative. This allows us to conclude that modifying a reversible MCMC method to improve its efficiency will also improve the efficiency of a method that randomly chooses either this or some other reversible method, and to conclude that improving the efficiency of a reversible update for one component of state (as in Gibbs sampling) will improve the overall efficiency of a reversible method that combines this and other updates. It also explains how antithetic MCMC can be more efficient than i.i.d. sampling. We also establish conditions that can guarantee that a method is not efficiency-dominated by any other method.
Paper Structure (8 sections, 24 theorems, 51 equations)

This paper contains 8 sections, 24 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Background Preliminaries
  3. Relating Asymptotic Variance to Eigenvalues
  4. Efficiency Dominance Equivalences
  5. Efficiency Dominance of Combined Chains
  6. Efficiency Dominance and Eigenvalues
  7. Re-deriving Peskun's Theorem
  8. Elementary Proof of Lemma \ref{['effequivlemma']}

Key Result

Lemma 1

If $P$ is reversible with respect to $\pi$ then: (a) the eigenvalues of $P$ are real; (b) these eigenvalues can be associated with real eigenvectors; (c) if $\lambda_i$ and $\lambda_j$ are eigenvalues of $P$ with $\lambda_i \ne \lambda_j$, and $v_i$ and $v_j$ are real eigenvectors associated with $\

Theorems & Definitions (24)

  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • Theorem 7
  • Lemma 8
  • Theorem 9
  • Theorem 10
  • ...and 14 more