Posterior propriety for hierarchical models with loglikelihoods that have norm bounds
Statisticians often use improper priors to express ignorance or to provide good frequency properties, requiring that posterior propriety be verified. Our paper addresses generalized linear mixed models, GLMMs, when Level I parameters have Normal distributions, with many commonlyused hyperpriors. It provides easytoverify sufficient posterior propriety conditions based on dimensions, matrix ranks, and exponentiated norm bounds, ENBs, for the Level I likelihood. Since many familiar likelihoods have ENBs, which is often verifiable via logconcavity and MLE finiteness, our novel use of ENBs permits unification of posterior propriety results and posterior MGF/moment results for many useful Level I distributions, including those commonly used with multilevel generalized linear models, e.g., GLMMs and hierarchical generalized linear models, HGLMs. Furthermore, those who need to verify existence of posterior distributions or of posterior MGFs/moments for a multilevel generalized linear model given a proper or improper multivariate F prior as in Section 1 should find the required results in Sections 1 and 2 and Theorem 3 (GLMMs), Theorem 4 (HGLMs), or Theorem 5 (posterior MGFs/moments).
 Authors:

^{[1]};
^{[2]}
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Harvard Univ., Cambridge, MA (United States)
 Publication Date:
 Report Number(s):
 LAUR1327701
Journal ID: ISSN 19316690
 Grant/Contract Number:
 AC5206NA25396
 Type:
 Accepted Manuscript
 Journal Name:
 Bayesian Analysis
 Additional Journal Information:
 Journal Volume: 10; Journal ID: ISSN 19316690
 Research Org:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; exponentiated norm bound; generalized linear mixed model; hierarchical generalized linear model; improper prior; multilevel objective Bayes
 OSTI Identifier:
 1235663
Michalak, Sarah E., and Morris, Carl N.. Posterior propriety for hierarchical models with loglikelihoods that have norm bounds. United States: N. p.,
Web. doi:10.1214/15BA962.
Michalak, Sarah E., & Morris, Carl N.. Posterior propriety for hierarchical models with loglikelihoods that have norm bounds. United States. doi:10.1214/15BA962.
Michalak, Sarah E., and Morris, Carl N.. 2015.
"Posterior propriety for hierarchical models with loglikelihoods that have norm bounds". United States.
doi:10.1214/15BA962. https://www.osti.gov/servlets/purl/1235663.
@article{osti_1235663,
title = {Posterior propriety for hierarchical models with loglikelihoods that have norm bounds},
author = {Michalak, Sarah E. and Morris, Carl N.},
abstractNote = {Statisticians often use improper priors to express ignorance or to provide good frequency properties, requiring that posterior propriety be verified. Our paper addresses generalized linear mixed models, GLMMs, when Level I parameters have Normal distributions, with many commonlyused hyperpriors. It provides easytoverify sufficient posterior propriety conditions based on dimensions, matrix ranks, and exponentiated norm bounds, ENBs, for the Level I likelihood. Since many familiar likelihoods have ENBs, which is often verifiable via logconcavity and MLE finiteness, our novel use of ENBs permits unification of posterior propriety results and posterior MGF/moment results for many useful Level I distributions, including those commonly used with multilevel generalized linear models, e.g., GLMMs and hierarchical generalized linear models, HGLMs. Furthermore, those who need to verify existence of posterior distributions or of posterior MGFs/moments for a multilevel generalized linear model given a proper or improper multivariate F prior as in Section 1 should find the required results in Sections 1 and 2 and Theorem 3 (GLMMs), Theorem 4 (HGLMs), or Theorem 5 (posterior MGFs/moments).},
doi = {10.1214/15BA962},
journal = {Bayesian Analysis},
number = ,
volume = 10,
place = {United States},
year = {2015},
month = {7}
}