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Likelihood Inference and Bayesian MCMC

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Much of Bayesian MCMC aims to be “objective” and hence is often strongly related to likelihood inference. A primary distinction between the two approaches involves whether nuisance parameters are maximized or integrated out of likelihood functions. This distinction has also been of interest in frequentist higher-order likelihood theory. In particular, there are various forms of adjustments to profile likelihood, aimed at reducing undesirable effects of fitting nuisance parameters. These adjustments are closely related to the distinction between profile likelihood and marginal posterior distributions under diffuse priors. We have explored using MCMC posterior samples on nuisance parameters to adjust posterior distributions for approximating profile likelihood, which differs from using special non-Bayesian MCMC methods to calculate the likelihood. However, the best of frequentist modified profile likelihoods can improve on the profile, and we also explore using MCMC to approximate these. This involves representing nuisance parameters orthogonally to interest parameters, in a sense stronger than usual. These issues seem also related to marginalization difficulties in fiducial inference.

Donald A. Pierce, Oregon Health & Science University Ruggero Bellio, Udine University, Italy

This talk is part of the Statistics series.

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