Data Augmentation Algorithm Sample Clauses

Data Augmentation Algorithm. In Bayesian Modeling, one generally wants to sample from a posterior den- sity. However, in real world applications, missing or unobserved data often prevents direct sampling from this posterior. Xxxxxx and Xxxx (1987) de- veloped the Data Augmentation (DA) algorithm, a technique similar to the EM algorithm in that we impute missing data or unobserved information from observed data to estimate parameters from a posterior distribution. Assume data Y is observed and augmented by the latent variable Z. Y has a distribution depending on θ, and so our objective is to sample from the posterior density P (θ|Y ). However, we may only be able to sample from the augmented posterior P (θ|Y, Z) and the predictive distribution P (Z|Y, θ). The DA algorithm provides and iterative process for sampling from these two es- timable distributions to approximate the posterior for θ. The iterative process of the DA algorithm has two steps. The first step is the Imputation or I-Step, in which using the current estimate of θ(k), Z is generated from P (Z|Y, θ(k)). The imputed Z is then used to update the ap- proximation of P (θ|Y ) in the Posterior or P-step. These two steps are repeated until there is convergence of the posterior distribution approximation(Xxxxxx & Xxxx 1987).