  Approximate Bayesian computation (ABC) methods permit approximate inference
for intractable likelihoods when it is possible to simulate from the model.
However they perform poorly for high dimensional data, and in practice must
usually be used in conjunction with dimension reduction methods, resulting in a
loss of accuracy which is hard to quantify or control. We propose a new ABC
method for high dimensional data based on rare event methods which we refer to
as RE-ABC. This uses a latent variable representation of the model. For a given
parameter value, we estimate the probability of the rare event that the latent
variables correspond to data roughly consistent with the observations. This is
performed using sequential Monte Carlo and slice sampling to systematically
search the space of latent variables. In contrast standard ABC can be viewed as
using a more naive Monte Carlo estimate. We use our rare event probability
estimator as a likelihood estimate within the pseudo-marginal
Metropolis-Hastings algorithm for parameter inference.
  We provide asymptotics showing that RE-ABC has a lower computational cost for
high dimensional data than standard ABC methods. We also illustrate our
approach empirically, on a Gaussian distribution and an application in
infectious disease modelling.
