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Activity Number: 239
Type: Contributed
Date/Time: Monday, August 5, 2013 : 2:00 PM to 3:50 PM
Sponsor: Biometrics Section
Abstract - #309752
Title: Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models
Author(s): Lisa Henn*+ and James Hodges
Companies: University of Minnesota and Univ of Minnesota
Keywords: variance component ; random effects ; generalized linear model ; precision reparamerization ; diagonalizable covariance
Abstract:

Scattered reports of multiple maxima in posterior distributions, likelihoods and restricted likelihoods appear throughout the literature. This paper surveys existing literature and proposes a unifying framework for understanding multiple maxima. For those problems with covariance structures that are diagonalizable in a specific sense, the restricted likelihood can be viewed as a generalized linear model (GLM) with gamma errors, identity link, and a prior distribution on the error variance. The GLM portion of the restricted likelihood can be made to conflict with the portion of the restricted likelihood that functions like a prior distribution on the error variance, giving two local maxima in the restricted likelihood. Applying an explicit conjugate prior distribution to variance parameters permits a second local maximum in the marginal posterior distribution even if the likelihood contribution has a single maximum. Also, reparameterization from variance to precision can change the posterior modality; the converse also is true. Modelers should beware of these potential pitfalls when selecting prior distributions and when using peak-finding algorithms to estimate parameters.


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