Statistical Analysis with Missing Data., 3rd Edition

10Bayes and Multiple Imputation

10.1 Bayesian Iterative Simulation Methods

10.1.1 Data Augmentation

A useful alternative approach to maximum likelihood (ML), particularly when the sample size is small, is to include a reasonable prior distribution for the parameters and compute the posterior distribution of the parameters of interest. We have already been introduced to this approach in Section 6.1.4 with complete data, and with missingness in Sections 7.3 and 7.4.4, in the special case of multivariate normal data with a monotone missingness pattern.

The posterior distribution for a model with ignorable missingness is

(10.1)

where p(θ) is the prior distribution and f (Y₍₀₎ ∣ θ) is the density of the observed data Y₍₀₎. In the examples of Chapter 7, simulation from the posterior distribution could be accomplished without iteration. Specifically, the likelihood was factored into complete data components,

and, assuming that the parameters (φ₁,…, φ_Q) were also a priori independent, the posterior distribution was factored in an analogous way, with φ₁,…, φ_Q a posteriori independent. Consequently, draws of (φ₁,…, φ_Q) could be obtained directly from the factored complete-data posterior ...