Description:

Title: Quantifying Nuisance Parameter Effects in Likelihood and Bayesian Inference

Todd Kuffner
Assistant Professor, Department of Mathematics
Washington University in St. Louis

Abstract:
In the age of computer-aided statistical inference, the practitioner has at her disposal an arsenal of computational tools to perform inference for low-dimensional parameters of interest, where the elimination of nuisance parameters can be accomplished by optimization, numerical integration, or some other computational sorcery. An operational assumption is that black-box tools can usually vaccinate the final inference for the interest parameter against potential effects arising from the presence of nuisance parameters. At the same time, from a theoretical, analytic perspective, accurate inference on a scalar interest parameter in the presence of nuisance parameters may be obtained by asymptotic refinement of likelihood-based statistics. Among these are Barndorff-Nielsen’s adjustment to the signed root likelihood ratio statistic, the Bartlett correction, and Cornish-Fisher transformations. We show how these adjustments may be decomposed into two terms, the first taking the same value when there are no nuisance parameters or there is an orthogonal nuisance parameter, and the second term being zero when there are no nuisance parameters. Illustrations are given for a number of examples which provide insight into the effect of nuisance parameters on parametric inference for parameters of interest. Connections and extensions for Bayesian inference are also discussed, and some open foundational questions are posed regarding the role of nuisance parameters in Bayesian inference, with some emphasis on possible effects in computational procedures for inference. Time permitting, I will explore potential links with recent work on post-selection inference.