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We consider a Bayesian hierarchical version of the normal theory general linear model which is practically relevant in the sense that it is general enough to have many applications and it is not straightforward to sample directly from the corresponding posterior distribution. Thus we study a block Gibbs sampler that has the posterior as its invariant distribution. In particular, we establish that the Gibbs sampler converges at a geometric rate. This allows us to establish conditions for a central limit theorem for the ergodic averages used to estimate features of the posterior. Geometric ergodicity is also a key component for using batch means methods to consistently estimate the variance of the asymptotic normal distribution. Together, our results give practitioners the tools to be as confident in inferences based on the observations from the Gibbs sampler as they would be with inferences based on random samples from the posterior. Our theoretical results are illustrated with an application to data on the cost of health plans issued by health maintenance organizations.
Exact inference for hidden Markov models requires the evaluation of all distributions of interest - filtering, prediction, smoothing and likelihood - with a finite computational effort. This article provides sufficient conditions for exact inference
Nested sampling is a simulation method for approximating marginal likelihoods proposed by Skilling (2006). We establish that nested sampling has an approximation error that vanishes at the standard Monte Carlo rate and that this error is asymptotical
Among Monte Carlo techniques, the importance sampling requires fine tuning of a proposal distribution, which is now fluently resolved through iterative schemes. The Adaptive Multiple Importance Sampling (AMIS) of Cornuet et al. (2012) provides a sign
In this paper, we analyze the convergence rate of a collapsed Gibbs sampler for crossed random effects models. Our results apply to a substantially larger range of models than previous works, including models that incorporate missingness mechanism an
This note presents a simple and elegant sampler which could be used as an alternative to the reversible jump MCMC methodology.