Abstract: The Empirical Bayes approach to high dimensional estimation or multiple testing assumes a model of sampled random variables Xi's with densities p(x/ti) of same form but different parameters ti, i=1,2,..n, and treats the ti's as iid with an unknown density f(t).Then Xi's are iid with density p(x/f)= integral of p(x/t)f(t).

A good estimate of f based on the Xi's is essential in making inference on the parameters ti.Once f is r=estimated it is relatively straightforward to make inference about the ti's using Xi and the estimate of f. This important methodology is very popular and has been applied to such problems as micro-arrays for gene expression and protein identification. In the gene expression problem, ti=0 corresponds to no effect of the ith gene on the corresponding variable Xi. If ti is not equal to 0, then the gene has an effect or is expressed.This is a testing problem, with the ti's iid with density f, with respect to a dominating measure which is the Lebesgue measure plus a point mass at zero.There are similar problems of estimating theta when the dominating measure is the Lebesgue measure. In applications ti's constitute tens of thousands of parameters and we test or estimate all the ti's.

Newton's algorithm, first proposed in 1998, provides a very fast algorithm, faster by an order of magnitude than NP Bayes estimates or NPMLE. However very little is known about its theoretical properties or intuitive motivation.Newton proved his algorithm converges but was unable to identify the limit as the true mixing distribution f.Moreover his proof has a serious gap.

This talk will provide an overview of the algorithm .I will present intuitive motivation, simulation and some rigorously proved consistency theorems.

The talk will be self contained, no background in Statistics is required.

It is based on joint work with Surya T. Tokdar of CMU and Ryan G. Martin of Purdue.