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2022 Sequentially Guided MCMC Proposals for Synthetic Likelihoods and Correlated Synthetic Likelihoods
Umberto Picchini, Umberto Simola, Jukka Corander
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Bayesian Anal. Advance Publication 1-31 (2022). DOI: 10.1214/22-BA1305

Abstract

Synthetic likelihood (SL) is a strategy for parameter inference when the likelihood function is analytically or computationally intractable. In SL, the likelihood function of the data is replaced by a multivariate Gaussian density over summary statistics of the data. SL requires simulation of many replicate datasets at every parameter value considered by a sampling algorithm, such as Markov chain Monte Carlo (MCMC), making the method computationally-intensive. We propose two strategies to alleviate the computational burden. First, we introduce an algorithm producing a proposal distribution that is sequentially tuned and made conditional to data, thus it rapidly guides the proposed parameters towards high posterior density regions. In our experiments, a small number of iterations of our algorithm is enough to rapidly locate high density regions, which we use to initialize one or several chains that make use of off-the-shelf adaptive MCMC methods. Our “guided” approach can also be potentially used with MCMC samplers for approximate Bayesian computation (ABC). Second, we exploit strategies borrowed from the correlated pseudo-marginal MCMC literature, to improve the chains mixing in a SL framework. Moreover, our methods enable inference for challenging case studies, when the posterior is multimodal and when the chain is initialised in low posterior probability regions of the parameter space, where standard samplers failed. To illustrate the advantages stemming from our framework we consider five benchmark examples, including estimation of parameters for a cosmological model and a stochastic model with highly non-Gaussian summary statistics.

Funding Statement

UP is supported by the Swedish Research Council (Vetenskapsrådet 2019-03924) and the Chalmers AI Research Centre (CHAIR). JC was funded by the ERC grant no. 742158. US was funded by Academy of Finland grant no. 320182.

Acknowledgments

We would like to thank Christopher Drovandi for useful feedback on an earlier draft of this paper. We also thank the anonymous reviewers for helping improve the paper, and especially the reviewer who suggested storing the sample mean of simulated summaries s¯ into D.

Citation

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Umberto Picchini. Umberto Simola. Jukka Corander. "Sequentially Guided MCMC Proposals for Synthetic Likelihoods and Correlated Synthetic Likelihoods." Bayesian Anal. Advance Publication 1 - 31, 2022. https://doi.org/10.1214/22-BA1305

Information

Published: 2022
First available in Project Euclid: 8 February 2022

arXiv: 2004.04558
Digital Object Identifier: 10.1214/22-BA1305

Keywords: Bayesian inference , cosmological parameters , intractable likelihoods , likelihood-free

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