Publication record · 18.cifr/1990.gelfand.gibbs-marginal

Sampling-Based Approaches to Calculating Marginal Densities

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Alan E. Gelfand (University of Connecticut), Adrian F. M. Smith (University of Nottingham)

RAI18.cifr/1990.gelfand.gibbs-marginal

Journal of the American Statistical Association· 1990· doi:10.1080/01621459.1990.10476213

We investigate the use of sampling-based methods for computing marginal densities in hierarchical Bayesian models. Gibbs sampling and related iterative conditional sampling schemes are shown to provide practical and general approximations to marginal posterior distributions. Rao-Blackwell estimators are introduced to reduce variance relative to direct Monte Carlo estimates.

Gibbs samplerBayesian computationMCMCmarginal densitiesRao-Blackwellhierarchical models

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High-dimensional Bayesian posterior integration was analytically intractable and numerical quadrature scaled poorly with parameter dimension. Applied Bayesians had no reliable general strategy for complex hierarchical models. Gelfand and Smith showed that Gibbs sampling provides a practical, scalable solution requiring only tractable conditional distributions.

Novelty

Gelfand and Smith (1990) unified and popularized the Gibbs sampler as a general-purpose Bayesian computation tool by demonstrating that iterative sampling from full conditional distributions converges to the correct joint posterior. They introduced Rao-Blackwellization as a variance-reduction technique that uses analytic conditionals rather than raw Monte Carlo averages.

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