Comparison of spline approximation with the modified likelihoods in the presence of nuisance parameter

Abstract

The likelihood functions from independent studies can be easily combined, and the combined likelihood function serves as a meaningful indication of the support the observed data give to the various parameter values. This fact has led many scientists to suggest using the likelihood function as a summary of post-data uncertainty concerning the parameter. Indeed, likelihood functions have several desired properties. They are objective, in that they depend only on the agreed-upon model and the data. They are also flexible, allowing us to combine information about competing models across studies. However, a serious difficulty arises because likelihood functions may not be expressible in a compact, easily-understood mathematical form suitable for communication or publication. For example, likelihood functions in mixture models may only be computable for individual values of the parameters and otherwise cannot be given in "closed form". To overcome this difficulty, we propose to approximate loglikelihood functions by using piecewise polynomials governed by a minimal number of parameters.