Sampling Based Approaches to Calculating Marginal Densities

Abstract
Stochastic substitution, the Gibbs sampler and the sampling-importance-resampling algorithm can be viewed as three alternative sampling, or Monte Carlo, based approaches to the calculation of numerical estimates of marginal probability distributions. The three approaches will be reviewed, and compared and contrasted, in relation to various joint probability structures frequently encountered in applications. In particular, the relevance of the approaches to calculating Bayesian posterior densities for a variety of structured models will be discussed and illustrated. Keywords: Marginal density; Monte Carlo sampling; Stochastic substitution; Gibbs sampler; Importance sampling; Conditional probability structure; Posterior distributions; Data augmentation; Hierarchical models; Missing data; Variance components.

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