  Approximate Bayesian computation (ABC) and synthetic likelihood (SL)
techniques have enabled the use of Bayesian inference for models that may be
simulated, but for which the likelihood cannot be evaluated 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 SL 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 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.
