  This is an up-to-date introduction to, and overview of, marginal likelihood
computation for model selection and hypothesis testing. Computing normalizing
constants of probability models (or ratio of constants) is a fundamental issue
in many applications in statistics, applied mathematics, signal processing and
machine learning. This article provides a comprehensive study of the
state-of-the-art of the topic. We highlight limitations, benefits, connections
and differences among the different techniques. Problems and possible solutions
with the use of improper priors are also described. Some of the most relevant
methodologies are compared through theoretical comparisons and numerical
experiments.
