Warped Gradient-Enhanced Gaussian Process Surrogate Models for Inference with Intractable Likelihoods

Markov chain Monte Carlo methods for intractable likelihoods, such as the exchange algorithm, require simulations of the sufficient statistics at every iteration of the Markov chain, which often result in expensive computations. Surrogate models for the likelihood function have been developed to accelerate inference algorithms in this context. However, these surrogate models tend to be relatively inflexible, and often provide a poor approximation to the true likelihood function. In this article, we propose the use of a warped, gradient-enhanced, Gaussian process surrogate model for the likelihood function, which jointly models the sample means and variances of the sufficient statistics, and uses warping functions to capture covariance nonstationarity in the input parameter space. We show that both the consideration of nonstationarity and the inclusion of gradient information can be leveraged to obtain a surrogate model that outperforms the conventional stationary Gaussian process surrogate model when making inference, particularly in regions where the likelihood function exhibits a phase transition. We also show that the proposed surrogate model can be used to improve the effective sample size per unit time when embedded in exact inferential algorithms. The utility of our approach in speeding up inferential algorithms is demonstrated on simulated and real-world data.