Tierney (1994) Markov Chains for Exploring Posterior Distributions

Summary

Tierney (1994) provides a unified mathematical treatment of Markov chain Monte Carlo (MCMC) methods for exploring posterior distributions, covering the Gibbs sampler, Metropolis-Hastings (MH) algorithm, and hybrid combinations. The paper draws on general state space Markov chain theory (Nummelin 1984) to derive convergence conditions, laws of large numbers, and central limit theorem (CLT) results for MCMC estimates, and applies standard simulation methodology (variance reduction via conditioning, batch means, sample-size guidance) to practical Bayesian computation.

Key Claims

The Hastings acceptance ratio $\alpha(x,y) = \min\{\pi(y)q(y,x)/[\pi(x)q(x,y)], 1\}$ is optimal over alternative forms (including Barker's 1965 form) by Peskun's (1973) theorem: it rejects candidate steps least often, minimising the asymptotic variance of MCMC estimates for any reversible chain.
Six types of Metropolis kernel are unified under a single framework: random walk, independence chain, rejection-sampling (RS) chain (embeds RS in MH to guarantee exactly $\pi$ even when the envelope fails), autoregressive/shrinkage, grid-based (Griddy-Gibbs embedded in MH), and reflection (induces negative correlations for variance reduction).
Irreducibility and aperiodicity are necessary and sufficient for convergence to the unique equilibrium $\pi$ (Theorem 1). Geometric ergodicity — verified via the drift condition $PV(x) \leq \lambda V(x) + b\cdot\mathbf{1}_C(x)$ with $\lambda < 1$ , $C$ small — implies a CLT for all $L^{2+\varepsilon}(\pi)$ functions (Theorem 4). Uniform ergodicity implies a CLT for all $L^2(\pi)$ functions (Theorem 5).
The asymptotic variance of a MCMC mean estimator is $\sigma^2 = \mathrm{Var}_\pi(g) + 2\sum_{j=1}^\infty \mathrm{Cov}(g(X_0), g(X_j))$ — the spectral density at frequency zero. Geometric ergodicity guarantees this sum converges absolutely.
Variance reduction via conditioning (Rao-Blackwellisation): replacing the raw MCMC average $\hat\mu = k^{-1}\sum g(X_j)$ with the conditional-expectation average $\hat\mu^c = k^{-1}\sum E[g(X_j)|Y_j]$ always reduces variance. In the nuclear pump example (Gelfand-Smith 1990), a 5-point sliding lattice conditioning kernel reduces estimator variance by a factor of 10 or more.
Hybrid algorithms (mixtures and cycles of kernels) provide a general mechanism for combining conjugate Gibbs steps with Metropolis steps where full conditionals are non-standard. Adding an occasional independence-chain step to a Gibbs sampler can restart mixing and reduce correlations while preserving the invariant distribution exactly.

Concepts Introduced or Extended

Entities Mentioned

Luke Tierney

Quotes

"Results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods."

"For computing averages it is usually desirable to reduce correlations, perhaps even making them negative, in order to reduce variances. As a result, no single Markov chain method will dominate all others in all problems."

My Take

This is the standard mathematical reference for the MH algorithm and MCMC convergence theory. The unification of six kernel types under a single Hastings framework and the Peskun optimality proof are the most durable theoretical contributions. The drift-condition approach to verifying geometric ergodicity (and hence CLT validity) is the tool practitioners use to justify inferential validity of MCMC estimates. The variance reduction via conditioning section connects MCMC to classical simulation efficiency ideas (Rao-Blackwell, antithetic variates) and foreshadows the multi-move and marginalisation strategies used in later Stochastic Volatility (SV) and Time-Varying Parameter VAR (TVP-VAR) samplers. The paper is more mathematical than computational: readers looking for worked Bayesian examples should complement it with Casella-George (1992) or Jackman (2000).