# Parallel Gaussian process surrogate Bayesian inference with noisy likelihood evaluations

## TL;DR

This paper introduces a parallel Bayesian inference method using Gaussian process surrogates to efficiently handle noisy likelihood evaluations, especially in complex simulator-based models, by enabling batch evaluations and sequential design.

## Contribution

It develops a novel batch-sequential design strategy for Gaussian process surrogates in Bayesian inference with noisy likelihoods, allowing parallel and sample-efficient evaluations.

## Key findings

- Method is robust across toy and simulation models.
- Enables highly parallelizable inference with fewer evaluations.
- Shows theoretical and empirical advantages over existing approaches.

## Abstract

We consider Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained. This occurs for example when complex simulator-based statistical models are fitted to data, and synthetic likelihood (SL) method is used to form the noisy log-likelihood estimates using computationally costly forward simulations. We frame the inference task as a sequential Bayesian experimental design problem, where the log-likelihood function is modelled with a hierarchical Gaussian process (GP) surrogate model, which is used to efficiently select additional log-likelihood evaluation locations. Motivated by recent progress in the related problem of batch Bayesian optimisation, we develop various batch-sequential design strategies which allow to run some of the potentially costly simulations in parallel. We analyse the properties of the resulting method theoretically and empirically. Experiments with several toy problems and simulation models suggest that our method is robust, highly parallelisable, and sample-efficient.