  We consider the problem of parametric statistical inference when likelihood
computations are prohibitively expensive but sampling from the model is
possible. Several so-called likelihood-free methods have been developed to
perform inference in the absence of a likelihood function. The popular
synthetic likelihood approach infers the parameters by modelling summary
statistics of the data by a Gaussian probability distribution. In another
popular approach called approximate Bayesian computation, the inference is
performed by identifying parameter values for which the summary statistics of
the simulated data are close to those of the observed data. Synthetic
likelihood is easier to use as no measure of `closeness' is required but the
Gaussianity assumption is often limiting. Moreover, both approaches require
judiciously chosen summary statistics. We here present an alternative inference
approach that is as easy to use as synthetic likelihood but not as restricted
in its assumptions, and that, in a natural way, enables automatic selection of
relevant summary statistic from a large set of candidates. The basic idea is to
frame the problem of estimating the posterior as a problem of estimating the
ratio between the data generating distribution and the marginal distribution.
This problem can be solved by logistic regression, and including regularising
penalty terms enables automatic selection of the summary statistics relevant to
the inference task. We illustrate the general theory on canonical examples and
employ it to perform inference for challenging stochastic nonlinear dynamical
systems and high-dimensional summary statistics.
