Practical perfect sampling using composite bounding chains: the Dirichlet-multinomial model

Nathan M. Stein and Xiao-Li Meng

Biometrika, 2013, vol. 100, issue 4, 817-830

Abstract: A discrete data augmentation scheme together with two different parameterizations yields two Gibbs samplers for sampling from the posterior distribution of the hyperparameters of the Dirichlet-multinomial hierarchical model under a default prior distribution. The finite-state space nature of this data augmentation permits us to construct two perfect samplers using bounding chains that take advantage of monotonicity and anti-monotonicity in the target posterior distribution, but both are impractically slow. We demonstrate that a composite algorithm that strategically alternates between the two samplers' updates can be substantially faster than either individually. The speed gains come because the composite algorithm takes a divide-and-conquer approach in which one update quickly shrinks the bounding set for the augmented data, and the other update immediately coalesces on the parameter, once the augmented-data bounding set is a singleton. We theoretically bound the expected time until coalescence for the composite algorithm, and show via simulation that the theoretical bounds can be close to actual performance. Copyright 2013, Oxford University Press.

Date: 2013
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