Brazilian Journal of Probability and Statistics

The limiting distribution of the Gibbs sampler for the intrinsic conditional autoregressive model

Marco A. R. Ferreira

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Abstract

We study the limiting behavior of the one-at-a-time Gibbs sampler for the intrinsic conditional autoregressive model with centering on the fly. The intrinsic conditional autoregressive model is widely used as a prior for random effects in hierarchical models for spatial modeling. This model is defined by full conditional distributions that imply an improper joint “density” with a multivariate Gaussian kernel and a singular precision matrix. To guarantee propriety of the posterior distribution, usually at the end of each iteration of the Gibbs sampler the random effects are centered to sum to zero in what is widely known as centering on the fly. While this works well in practice, this informal computational way to recenter the random effects obscures their implied prior distribution and prevents the development of formal Bayesian procedures. Here we show that the implied prior distribution, that is, the limiting distribution of the one-at-a-time Gibbs sampler for the intrinsic conditional autoregressive model with centering on the fly is a singular Gaussian distribution with a covariance matrix that is the Moore–Penrose inverse of the precision matrix. This result has important implications for the development of formal Bayesian procedures such as reference priors and Bayes-factor-based model selection for spatial models.

Article information

Source
Braz. J. Probab. Stat., Volume 33, Number 4 (2019), 734-744.

Dates
Received: April 2018
Accepted: February 2019
First available in Project Euclid: 26 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1566806430

Digital Object Identifier
doi:10.1214/19-BJPS435

Mathematical Reviews number (MathSciNet)
MR3996314

Zentralblatt MATH identifier
07120731

Keywords
Areal data ICAR models Markov random fields spatial data

Citation

Ferreira, Marco A. R. The limiting distribution of the Gibbs sampler for the intrinsic conditional autoregressive model. Braz. J. Probab. Stat. 33 (2019), no. 4, 734--744. doi:10.1214/19-BJPS435. https://projecteuclid.org/euclid.bjps/1566806430


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