## George Casella, Cornell University

### Algorithms, Approximations and Inference

Many difficult
problems can be solved with algorithms (such as MCMC) and
approximations (such as saddlepoints). When using these computational
tools, we must not forget
- - The usefulness of statistical methodology
- - The validity of the resulting inference
- - The power of using all of our tools (Bayes, frequentist,
likelihood, etc.)
We give three examples of such problems and solutions. The first
involves hierarchical models and an empirical Bayes solution (using an
EM algorithm); the second problem involves elimination of nuisance
parameters, using Bayesian matching prior technology to get a
frequentist solution; and the third problem concerns trapping in Gibbs
sampling with mixture models, where a solution is obtained by
Rao-Blackwellization. In each situation, addressing the problem in a
natural way leads to a procedure, and an inference, that is a
synthesis of Bayesian and frequentist methods.