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Paper No. 09-37

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F Rigat and A Mira

Parallel hierarchical sampling: a general-purpose class of multiple-chains MCMC algorithms

Abstract: This paper introduces the Parallel Hierarchical Sampler (PHS), a class of Markov chain Monte Carlo algorithms using several interacting chains having the same target distribution but different mixing properties. Unlike any single-chain MCMC algorithm, upon reaching stationarity one of the PHS chains, which we call the “mother” chain, attains exact Monte Carlo sampling of the target distribution of interest. We empirically show that this translates in a dramatic improvement in the sampler’s performance with respect to single-chain MCMC algorithms. Convergence of the PHS joint transition kernel is proved and its relationships with single-chain samplers, Parallel Tempering (PT) and variable augmentation algorithms are discussed. We then provide two illustrative examples comparing the accuracy of PHS with that of various Metropolis-Hastings and PT for sampling multimodal mixtures of multivariate Gaussian densities and for ’banana-shaped’ multivariate distributions with heavy tails. Finally, PHS is applied to approximate inferences for two Bayesian model uncertainty problems, namely selection of main effects for a linear Gaussian multiple regression model and inference for the structure of an exponential treed survival model.

Keywords: Bayesian model selection, classification and regression trees, Gaussian mixtures, heavy tails, linear regression, Metropolis-Hastings algorithm, multimodality, multiple-chainsMarkov chainMonte Carlo methods, survival analysis.