Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models |
| |
Authors: | Lisa Henn James S. Hodges |
| |
Affiliation: | Division of Biostatistics, School of Public Health, University of Minnesota, , SE, Minneapolis, MN 55455, USA |
| |
Abstract: | Scattered reports of multiple maxima in posterior distributions or likelihoods for mixed linear models appear throughout the literature. Less scrutinised is the restricted likelihood, which is the posterior distribution for a specific prior distribution. This paper surveys existing literature and proposes a unifying framework for understanding multiple maxima. For those problems with covariance structures that are diagonalisable in a specific sense, the restricted likelihood can be viewed as a generalised linear model with gamma errors, identity link and a prior distribution on the error variance. The generalised linear model 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 in addition 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. Moreover, reparameterisation from variance to precision can change the posterior modality; the converse also is true. Modellers should beware of these potential pitfalls when selecting prior distributions or using peak‐finding algorithms to estimate parameters. |
| |
Keywords: | Variance component random effects generalised linear model precision reparameterisation diagonalisable covariance |
|
|