Stability of the Gibbs sampler for Bayesian hierarchical models

By Omiros Papaspiliopoulos, Gareth O. Roberts. 2007

59 citations recorded by the source corpus.

We characterize the convergence of the Gibbs sampler which samples from the joint posterior distribution of parameters and missing data in hierarchical linear models with arbitrary symmetric error distributions. We show that the convergence can be uniform, geometric or subgeometric depending on the relative tail behavior of the error distributions, and on the parametrization chosen. Our theory is applied to characterize the convergence of the Gibbs sampler on latent Gaussian process models. We indicate how the theoretical framework we introduce will be useful in analyzing more complex models.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cb5e615976c075d0d9814f0d145d03e771c6b466

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Citations in the corpus

Unbiased Markov chain Monte Carlo methods with couplings (2020, background): “The parameterization of a Gibbs sampler will also certainly impact not only its marginal performance (Papaspiliopoulos and Roberts, 2008) but also the implementation and performance of coupling strategies.”
A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants (2011, methodology): “Simple examples of subgeometric DA chains can be found in Papaspiliopoulos and Roberts (2008) and Tan (2008), Chapter 4. Condition (4) has appeared in the Markov chain Monte Carlo literature before. It is exactly the bivariate version of Liu, Wong and Kong’s (1995) “Condition (b)” and it was also em”
Adaptive Gibbs samplers and related MCMC methods (2011, background): “Such issues have been investigated in e.g. [38] and more recently in relation to the parametrization of hierarchical models (see [35] and references therein).”
A General Framework for the Parametrization of Hierarchical Models (2007, background): “Roberts and Stramer [38] show that the CA on ( X 1: n , Θ) is O ( n ); that is, the better we try to approximate the likelihood [and therefore the posterior P (Θ | Y )], the worse the convergence of the algorithm.”
To Center or Not to Center: That Is Not the Question—An Ancillarity–Sufficiency Interweaving Strategy (ASIS) for Boosting MCMC Efficiency (2011, methodology): “…the notions of SA and AA are mathematically equivalent to the centered parameterization (CP) and non-centered parameterization (NCP), respectively (see Gelfand, Sahu, and Carlin 1995, 1996; Roberts and Sahu 1997; Papaspiliopoulos, Roberts, and Skold 2003, 2007; Papaspiliopoulos and Roberts 2008).”
Analysis of the Gibbs Sampler for Hierarchical Inverse Problems (2013, methodology): “This is precisely the case with the Gibbs sampler and related MwG algorithms, which are particularly hard to analyze (see, for example, [32]).”