  Approximate Bayesian computation (ABC) is the most popular approach to
inferring parameters in the case where the data model is specified in the form
of a simulator. It is not possible to directly implement standard Monte Carlo
methods for inference in such a model, due to the likelihood not being
available to evaluate pointwise. The main idea of ABC is to perform inference
on an alternative model with an approximate likelihood (the ABC likelihood),
estimated at each iteration from points simulated from the data model. The
central challenge of ABC is then to trade-off bias (introduced by approximating
the model) with the variance introduced by estimating the ABC likelihood.
Stabilising the variance of the ABC likelihood requires a computational cost
that is exponential in the dimension of the data, thus the most common approach
to reducing variance is to perform inference conditional on summary statistics.
In this paper we introduce a new approach to estimating the ABC likelihood:
using iterative ensemble Kalman inversion (IEnKI) (Iglesias, 2016; Iglesias et
al., 2018). We first introduce new estimators of the marginal likelihood in the
case of a Gaussian data model using the IEnKI output, then show how this may be
used in ABC. Performance is illustrated on the Lotka-Volterra model, where we
observe substantial improvements over standard ABC and other commonly-used
approaches.
