  Many statistical models have likelihoods which are intractable: it is
impossible or too expensive to compute the likelihood exactly. In such
settings, a common approach is to replace the likelihood with an approximation,
and proceed with inference as if the approximate likelihood were the exact
likelihood. In this paper, we describe conditions on the approximate likelihood
which guarantee that this naive inference with an approximate likelihood has
the same first-order asymptotic properties as inference with the exact
likelihood. We investigate the implications of these results for inference
using a Laplace approximation to the likelihood in a simple two-level latent
variable model, and using reduced dependence approximations to the likelihood
in an Ising model on a lattice.
