Zero Variance Markov Chain Monte Carlo for Bayesian Estimators

Abstract

Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the idea is illustrated with real applications to probit, logit and GARCH Bayesian models. For all these models, a central limit theorem and unbiasedness for the zero-variance estimator are proved (see the supplementary material available on-line).

26 pages, 4 figures. This is an updated version: the results are the same as the previous one, but presentation is more essential

References in corpus (4)

• Zero-variance principle for Monte Carlo algorithms
• Zero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to Forces
• Multiprocess parallel antithetic coupling for backward and forward Markov Chain Monte Carlo
• Control Variates for Reversible MCMC Samplers

Cited by in corpus (19)

• A Conceptual Introduction to Hamiltonian Monte Carlo
• Accelerating pseudo-marginal Metropolis-Hastings by correlating auxiliary variables
• Post-Processing of MCMC
• Neural Control Variates for Variance Reduction
• Reduced variance analysis of molecular dynamics simulations by linear combination of estimators
• Diffusion approximations and control variates for MCMC
• The Controlled Thermodynamic Integral for Bayesian Model Comparison
• Scalable Control Variates for Monte Carlo Methods via Stochastic Optimization
• Control functionals for Monte Carlo integration
• Stochastic Stein Discrepancies
• Split Sampling: Expectations, Normalisation and Rare Events
• Vector-Valued Control Variates
• Fast compression of MCMC output
• Posterior Integration on a Riemannian Manifold
• Doubly Adaptive Importance Sampling
• Variance reduction for dependent sequences with applications to Stochastic Gradient MCMC
• Systematic lowering of the scaling of Monte Carlo calculations by partitioning andsubsampling
• Variance reduction for Markov chains with application to MCMC
• Discussion of "Unbiased Markov chain Monte Carlo with couplings" by Pierre E. Jacob, John O'Leary and Yves F. Atchadé