  Generalized linear models play an essential role in a wide variety of
statistical applications. This paper discusses an approximation of the
likelihood in these models that can greatly facilitate computation. The basic
idea is to replace a sum that appears in the exact log-likelihood by an
expectation over the model covariates; the resulting "expected log-likelihood"
can in many cases be computed significantly faster than the exact
log-likelihood. In many neuroscience experiments the distribution over model
covariates is controlled by the experimenter and the expected log-likelihood
approximation becomes particularly useful; for example, estimators based on
maximizing this expected log-likelihood (or a penalized version thereof) can
often be obtained with orders of magnitude computational savings compared to
the exact maximum likelihood estimators. A risk analysis establishes that these
maximum EL estimators often come with little cost in accuracy (and in some
cases even improved accuracy) compared to standard maximum likelihood
estimates. Finally, we find that these methods can significantly decrease the
computation time of marginal likelihood calculations for model selection and of
Markov chain Monte Carlo methods for sampling from the posterior parameter
distribution. We illustrate our results by applying these methods to a
computationally-challenging dataset of neural spike trains obtained via
large-scale multi-electrode recordings in the primate retina.
