Bootstrapped synthetic likelihood

The development of approximate Bayesian computation (ABC) and synthetic likelihood (SL) techniques has enabled the use of Bayesian inference for models that may be simulated, but for which the likelihood is not available to evaluate pointwise at values of an unknown parameter $\theta$. The main idea in ABC and SL is to, for different values of $\theta$ (usually chosen using a Monte Carlo algorithm), build estimates of the likelihood based on simulations from the model conditional on $\theta$. The quality of these estimates determines the efficiency of an ABC/SL algorithm. In standard ABC/SL, the only means to improve an estimated likelihood at $\theta$ is to simulate more times from the model conditional on $\theta$, which is infeasible in cases where the simulator is computationally expensive. In this paper we describe how to use bootstrapping as a means for improving synthetic likelihood estimates whilst using fewer simulations from the model, and also investigate its use in ABC. Further, we investigate the use of the bag of little bootstraps as a means for applying this approach to large datasets, yielding to Monte Carlo algorithms that accurately approximate posterior distributions whilst only simulating subsamples of the full data. Examples of the approach applied to i.i.d., temporal and spatial data are given.