The process of inferring conclusions from experimental observations often requires the construction and evaluation of a likelihood function, which gives the probability, or probability density, of the observed data given a particular statistical model.
The likelihood function forms the basis for many statistical inference techniques. For example, one could estimate the parameters of a model by maximizing the likelihood. The problem is that likelihood functions frequently contain more parameters than we care about. The inexplicable or uninteresting noise that buffets processes of interest has to be parameterized and accounted for in the likelihood function. As Berger, Liseo, and Wolpert discuss in this paper, the existence of so-called ‘nuisance parameters’ severely hampers inference in many cases. The authors review a few of the common frequentist techniques for dealing with nuisance parameters in likelihood functions, but fall strongly in favor of integrating the likelihood function over the nuisance parameters. Although this method has a Bayesian flavor to it, the authors emphasize the practical benefits of integrated likelihoods, even for statisticians with more frequentist leanings.
[This post previously appeared in klogw.org].