Efficient Acquisition Rules for Model-Based Approximate Bayesian Computation

Abstract

Approximate Bayesian computation (ABC) is a method for Bayesian inference when the likelihood is unavailable but simulating from the model is possible. However, many ABC algorithms require a large number of simulations, which can be costly. To reduce the computational cost, Bayesian optimisation (BO) and surrogate models such as Gaussian processes have been proposed. Bayesian optimisation enables one to intelligently decide where to evaluate the model next but common BO strategies are not designed for the goal of estimating the posterior distribution. Our paper addresses this gap in the literature. We propose to compute the uncertainty in the ABC posterior density, which is due to a lack of simulations to estimate this quantity accurately, and define a loss function that measures this uncertainty. We then propose to select the next evaluation location to minimise the expected loss. Experiments show that the proposed method often produces the most accurate approximations as compared to common BO strategies.

Bayesian emulator optimisation for cosmology: application to the Lyman-alpha forest

“…Therefore, this random displacement can be drawn from a Gaussian distribution with a full-width-at-half-maximum set by the current estimation of the posterior contours (most simply approximated by using the desired number (e. g., two) of sigma from the 1D marginalised distributions; see ref. [56] for discussion and comparison of deterministic and stochastic acquisition rules). In general, these "exploration" terms should be tuned to the size of the credible region which is required to be well estimated.…”

“…This naturally dovetails into Gaussian process emulation, which provides a robust estimate of uncertainty in predictions across the parameter space [51]. The details of the balance between exploration and exploitation are encoded in the acquisition function used to determine future proposals, of which many examples have been developed (we use a novel expansion of the GP-UCB acquisition function) [52][53][54][55][56]. We also show how to propose multiple training samples simultaneously (batch acquisition).…”

Bayesian optimization for likelihood-free cosmological inference

“…As pointed out by Järvenpää et al . (2017), in Bayesian optimisation for approximate Bayesian computation, the goal should not be to find the minimum of J(θ), but rather to minimise the expected uncertainty in the estimate of the approximate posterior over the future evaluation of the simulator at θ .…”

“…In high dimension, the integral can become prohibitively expensive to compute on a grid. As discussed by Järvenpää et al . (2017), it can then be evaluated with Monte Carlo or quasi-Monte Carlo methods such as importance sampling.…”

“…We focus on computable parametric appro-ximations to the true likelihood (also known as synthetic likelihoods), rendering the approach completely ε-free. Recently, Järvenpää et al . (2017) introduced an acquisition function for Bayesian optimisation (the expected integrated variance), specifically tailored to perform efficient and accurate ABC.…”