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We consider posterior sampling in the very common Bayesian hierarchical modelin which observed data depends on high-dimensional latent variables that, inturn, depend on relatively few hyperparameters. When the full conditional overthe latent variables has a known form, the marginal posterior distribution overhyperparameters is accessible and can be sampled using a Markov chain MonteCarlo (MCMC) method on a low-dimensional parameter space. This may improvecomputational efficiency over standard Gibbs sampling since computation is notover the high-dimensional space of latent variables and correlations betweenhyperparameters and latent variables become irrelevant. When the marginalposterior over hyperparameters depends on a fixed-dimensional sufficientstatistic, precomputation of the sufficient statistic renders the cost of thelow-dimensional MCMC independent of data size. Then, when the hyperparametersare the primary variables of interest, inference may be performed in big-datasettings at modest cost. Moreover, since the form of the full conditional forthe latent variables does not depend on the form of the hyperpriordistribution, the method imposes no restriction on the hyperprior, unlike Gibbssampling that typically requires conjugate distributions. We demonstrate theseefficiency gains in four computed examples.

Sampling hyperparameters in hierarchical models: improving on Gibbs for high-dimensional latent fields and large data sets