Posterior propriety for hierarchical models with log-likelihoods that have norm bounds

Michalak, Sarah E.; Morris, Carl N.

doi:10.1214/15-BA962

Title: Posterior propriety for hierarchical models with log-likelihoods that have norm bounds

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Abstract

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 commonly-used hyperpriors. It provides easy-to-verify 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 log-concavity 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:

Michalak, Sarah E. ^[1]; Morris, Carl N. ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Harvard Univ., Cambridge, MA (United States)

Publication Date:: Fri Jul 17 00:00:00 EDT 2015

Research Org.:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1235663

Report Number(s):: LA-UR-13-27701
Journal ID: ISSN 1931-6690

Grant/Contract Number:: AC52-06NA25396

Resource Type:: Accepted Manuscript

Journal Name:: Bayesian Analysis

Additional Journal Information:: Journal Volume: 10; Journal ID: ISSN 1931-6690

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

Citation Formats


                    Michalak, Sarah E., and Morris, Carl N. Posterior propriety for hierarchical models with log-likelihoods that have norm bounds.  United States: N. p., 2015. 
Web.  doi:10.1214/15-BA962.

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                    Michalak, Sarah E., & Morris, Carl N. Posterior propriety for hierarchical models with log-likelihoods that have norm bounds.  United States.  https://doi.org/10.1214/15-BA962

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                    Michalak, Sarah E., and Morris, Carl N. Fri .  
"Posterior propriety for hierarchical models with log-likelihoods that have norm bounds".  United States.  https://doi.org/10.1214/15-BA962.  https://www.osti.gov/servlets/purl/1235663.

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@article{osti_1235663,

  title        = {Posterior propriety for hierarchical models with log-likelihoods 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 commonly-used hyperpriors. It provides easy-to-verify 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 log-concavity 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          = {10.1214/15-BA962},

  journal      = {Bayesian Analysis},

  number       = ,

  volume       = 10,

  place        = {United States},

  year         = {Fri Jul 17 00:00:00 EDT 2015},

  month        = {Fri Jul 17 00:00:00 EDT 2015}

}

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