Many models of interest in the natural and social sciences have no
closed-form likelihood function, which means that they cannot be
treated using the usual techniques of statistical inference. In the
case where such models can be efficiently simulated, Bayesian
inference is still possible thanks to the Approximate Bayesian
Computation (ABC) algorithm. Although many refinements have since been
suggested, the technique suffers from three major shortcomings. First,
it requires introducing a vector of "summary statistics", the choice
of which is arbitrary and may lead to strong biases. Second, ABC may
be excruciatingly slow due to very low acceptance rates. Third, it
cannot produce a reliable estimate of the marginal likelihood of the
model.
We introduce a technique that solves the first and the third
issues, and considerably alleviates the second. We adapt to the
likelihood-free context a variational approximation algorithm,
Expectation Propagation (Minka, 2001). The resulting algorithm is
shown to be faster by a few orders of magnitude than alternative
algorithms, while producing an overall approximation error which is
typically negligible. Comparisons are performed in three real-world
applications which are typical of likelihood-free inference, including
one application in neuroscience which is novel, and possibly too
challenging for standard ABC techniques.