Abstract
Performing numerical integration when the integrand itself cannot be evaluated point-wise is a challenging task that arises in statistical analysis, notably in Bayesian inference for models with intractable likelihood functions. Markov chain Monte Carlo (MCMC) algorithms have been proposed for this setting, such as the pseudo-marginal method for latent variable models and the exchange algorithm for a class of undirected graphical models. As with any MCMC algorithm, the resulting estimators are justified asymptotically in the limit of the number of iterations, but exhibit a bias for any fixed number of iterations due to the Markov chains starting outside of stationarity. This “burn-in” bias is known to complicate the use of parallel processors for MCMC computations. We show how to use coupling techniques to generate unbiased estimators in finite time, building on recent advances for generic MCMC algorithms. We establish the theoretical validity of some of these procedures, by extending existing results to cover the case of polynomially ergodic Markov chains. The efficiency of the proposed estimators is compared with that of standard MCMC estimators, with theoretical arguments and numerical experiments including state space models and Ising models.
Highlights
For various statistical models the likelihood function cannot be computed point-wise, which prevents the use of standard Markov chain Monte Carlo (MCMC) algorithms such as Metropolis–Hastings (MH) for Bayesian inference
The likelihood of latent variable models typically involves an intractable integral over the latent space. One can address this problem by designing MCMC algorithms on the joint space of parameters and latent variables
The following demonstrates the proposed algorithm for a latent variable model applied to polling data and explores the possible gains when compared to the unbiased estimators obtained using the coupled pseudo-marginal algorithm
Summary
For various statistical models the likelihood function cannot be computed point-wise, which prevents the use of standard Markov chain Monte Carlo (MCMC) algorithms such as Metropolis–Hastings (MH) for Bayesian inference. Pseudo-marginal methods have been proposed for these situations [Lin et al, 2000, Beaumont, 2003, Andrieu and Roberts, 2009], whereby unbiased Monte Carlo estimators of the likelihood are used within an MH acceptance mechanism while still producing chains that are ergodic with respect to the exact posterior distribution of interest, denoted by π. A method has been proposed to completely remove the bias of Markov chain ergodic averages [Glynn and Rhee, 2014] An extension of this approach using coupling ideas was proposed by Jacob et al [2020] and applied to a variety of MCMC algorithms. This methodology involves the construction of a pair of Markov chains, which are simulated until an event occurs. Accompanying code used for simulations and to generate the figures are provided at https://github.com/ lolmid/unbiased_intractable_targets
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