[1] |
Omiros Papaspiliopoulos and Gareth Roberts.
Stability of the Gibbs sampler for Bayesian hierarchical models.
Ann. Statist., 36(1):95-117, 2008.
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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. Keywords: Geometric ergodicity; capacitance; collapsed Gibbs sampler; state-space models; parametrization; Bayesian robustness |

[2] |
Omiros Papaspiliopoulos and Matteo Ruggiero.
Optimal filtering and the dual process.
Bernoulli, to appear, 2013.
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We link optimal filtering for hidden Markov models to the notion of duality for Markov processes. We show that when the signal is dual to a Êprocess that has two components, one deterministic and one a pure death process, and with respect to functions that define changes of measure conjugate to the emission density, the filtering distributions evolve in the family of finite mixtures of such measures and the filter can be computed at a cost that is polynomial in the number of observations. Hence, for models in this framework, optimal filtering reduces to a version of the Baum-Welch filter. Special cases of our framework are the Kalman filter, but also models where the signal is the Cox-Ingersoll-Ross process and the one-dimensional Wright-Fisher process, which have been investigated before in the literature. The duals of these two processes that Êwe identify in this paper appear to be new in the literature. We also discuss the extensions of these results to an infinite-dimensional signal modelled as a Fleming-Viot process, and the connection of the duality framework we develop here and Kingman's coalescent. Keywords: Auxiliary variables;Bayesian conjugacy;Dirichlet process;Finite mixture models;Cox-Ingersoll-Ross process;Hidden Markov model;Kalman filters |

[3] |
Sergios Agapiou, Johnathan Bardsley, Omiros Papaspiliopoulos, and Andrew M.
Stuart.
Analysis of the Gibbs sampler for hierarchical inverse problems.
submitted, 2013.
[ bib |
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Many inverse problems arising in applications come from continuum models where the unknown parameter is a field. In practice the unknown field is discretized resulting in a problem in Keywords: Gaussian process priors, Markov chain Monte Carlo, inverse covariance operators, hierarchical models, diffusion limit |

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