Propriety of Posteriors in Structured Additive Regression Models: Theory and Empirical Evidence

Fahrmeir, Ludwig and Kneib, Thomas (2006): Propriety of Posteriors in Structured Additive Regression Models: Theory and Empirical Evidence. Collaborative Research Center 386, Discussion Paper 510

Abstract

Structured additive regression comprises many semiparametric regression models such as generalized additive (mixed) models, geoadditive models, and hazard regression models within a unified framework. In a Bayesian formulation, nonparametric functions, spatial effects and further model components are specified in terms of multivariate Gaussian priors for high-dimensional vectors of regression coefficients. For several model terms, such as penalised splines or Markov random fields, these Gaussian prior distributions involve rank-deficient precision matrices, yielding partially improper priors. Moreover, hyperpriors for the variances (corresponding to inverse smoothing parameters) may also be specified as improper, e.g. corresponding to Jeffery's prior or a flat prior for the standard deviation. Hence, propriety of the joint posterior is a crucial issue for full Bayesian inference in particular if based on Markov chain Monte Carlo simulations. We establish theoretical results providing sufficient (and sometimes necessary) conditions for propriety and provide empirical evidence through several accompanying simulation studies.

• Item Type: Paper (Research Paper)
• Subjects: Mathematics, Computer Science and Statistics; Mathematics, Computer Science and Statistics > Statistics; Mathematics, Computer Science and Statistics > Statistics > Collaborative Research Center 386
• Dewey Classification: 600 Natural sciences and mathematics; 600 Natural sciences and mathematics > 510 Mathematics