Variance Reduction for the Independent Metropolis Sampler
This addresses variance reduction in MCMC sampling for Bayesian inference, offering incremental improvements for computational statistics.
The paper tackles variance reduction for the Independent Metropolis sampler by proving that using a proposal density close to the target under KL divergence, combined with control variates, yields smaller asymptotic variance than i.i.d. sampling, with no extra computational cost but requiring analytical expectations under the proposal. It demonstrates this in linear regression with prior-likelihood conflict and proposes an adaptive algorithm for Bayesian logistic and Gaussian process regression.
Assume that we would like to estimate the expected value of a function $F$ with respect to an intractable density $π$, which is specified up to some unknown normalising constant. We prove that if $π$ is close enough under KL divergence to another density $q$, an independent Metropolis sampler estimator that obtains samples from $π$ with proposal density $q$, enriched with a variance reduction computational strategy based on control variates, achieves smaller asymptotic variance than i.i.d.\ sampling from $π$. The control variates construction requires no extra computational effort but assumes that the expected value of $F$ under $q$ is analytically available. We illustrate this result by calculating the marginal likelihood in a linear regression model with prior-likelihood conflict and a non-conjugate prior. Furthermore, we propose an adaptive independent Metropolis algorithm that adapts the proposal density such that its KL divergence with the target is being reduced. We demonstrate its applicability in a Bayesian logistic and Gaussian process regression problems and we rigorously justify our asymptotic arguments under easily verifiable and essentially minimal conditions.