Nesting EM algorithms for computational efficiency

Computing posterior modes (e.g., maximum likelihood estimates) for models i nvolving latent variables or missing data often involves complicated optimi zation procedures. By splitting this task into two simpler parts, however, EM-type algorithms often offer a simple solution. Although this approach ha s proven useful, in some settings even these simpler tasks are challenging. In particular, computations involving latent variables are typically diffi cult to simplify;. Thus, in models such as hierarchical models with complic ated latent variable structures, computationally intensive methods may be r equired for the expectation step of EM. This paper describes how nesting tw o or more EM algorithms can take advantage of closed form conditional expec tations and lead to algorithms which converge faster, are straightforward t o implement, and enjoy stable convergence properties. Methodology to monito r convergence of nested EM, algorithms is developed using importance and br idge sampling. The strategy is applied to hierarchical probit and t regress ion models to derive algorithms which incorporate aspects of Monte-Carlo EM , PX-EM, and nesting in order to combine computational efficiency with easy implementation.