Accelerating pseudo-marginal MCMC using Gaussian processes

Abstract

The grouped independence Metropolis-Hastings (GIMH) and Markov chain within Metropolis (MCWM) algorithms are pseudo-marginal methods used to perform Bayesian inference in latent variable models. These methods replace intractable likelihood calculations with unbiased estimates within Markov chain Monte Carlo algorithms. The GIMH method has the posterior of interest as its limiting distribution, but suffers from poor mixing if it is too computationally intensive to obtain high-precision likelihood estimates. The MCWM algorithm has better mixing properties, but tends to give conservative approximations of the posterior and is still expensive. A new method is developed to accelerate the GIMH method by using a Gaussian process (GP) approximation to the log-likelihood and train this GP using a short pilot run of the MCWM algorithm. This new method called GP-GIMH is illustrated on simulated data from a stochastic volatility and a gene network model. The new approach produces reasonable posterior approximations in these examples with at least an order of magnitude improvement in computing time. Code to implement the method for the gene network example can be found at http://www.runmycode.org/companion/view/2663.

A Function Emulation Approach for Doubly Intractable Distributions

“…Although delayed-acceptance MCMC approaches are asymptotically exact, the efficiency gains are limited because the methods require evaluating expensive likelihood functions at the second stage for promising proposals. Drovandi et al (2018) proposes an approach to speed up pseudo-marginal methods by replacing the log of an unbiased likelihood estimate with a Gaussian process approximation. Our approach has similarities to Drovandi et al (2018) in that we replace the log of the function estimate with a Gaussian process approximation, and also use a short run of an MCMC algorithm to obtain good design points for constructing the Gaussian process approximation.…”

“…Drovandi et al (2018) proposes an approach to speed up pseudo-marginal methods by replacing the log of an unbiased likelihood estimate with a Gaussian process approximation. Our approach has similarities to Drovandi et al (2018) in that we replace the log of the function estimate with a Gaussian process approximation, and also use a short run of an MCMC algorithm to obtain good design points for constructing the Gaussian process approximation. Function emulation approaches such as Drovandi et al (2018) and the method we describe in this manuscript, are useful for problems where it is expensive to evaluate a function (or evaluate a function approximation) many times.…”

“…Our approach has similarities to Drovandi et al (2018) in that we replace the log of the function estimate with a Gaussian process approximation, and also use a short run of an MCMC algorithm to obtain good design points for constructing the Gaussian process approximation. Function emulation approaches such as Drovandi et al (2018) and the method we describe in this manuscript, are useful for problems where it is expensive to evaluate a function (or evaluate a function approximation) many times. Both our algorithm and the algorithms in Drovandi et al (2018) require a pre-computation step which itself involves repeated approximations of the likelihood function at a set of parameter values.…”

“…Function emulation approaches such as Drovandi et al (2018) and the method we describe in this manuscript, are useful for problems where it is expensive to evaluate a function (or evaluate a function approximation) many times. Both our algorithm and the algorithms in Drovandi et al (2018) require a pre-computation step which itself involves repeated approximations of the likelihood function at a set of parameter values. Drovandi et al (2018) requires unbiased likelihood estimates in the pre-computation step, which can be prohibitively expensive for the problems we consider -interaction point processes and exponential random graph models with large data sets.…”

“…Both our algorithm and the algorithms in Drovandi et al (2018) require a pre-computation step which itself involves repeated approximations of the likelihood function at a set of parameter values. Drovandi et al (2018) requires unbiased likelihood estimates in the pre-computation step, which can be prohibitively expensive for the problems we consider -interaction point processes and exponential random graph models with large data sets. In contrast, our approach, as we demonstrate later in the paper, is computationally efficient even for such problems.…”