Two Groups and One Group Models
for Multiple Tests for Microarrays and Other Examples...a Survey and New
Results
I will first present a survey based on the papers of Efron,
Storey, Genovese and Wasserman, Scott and Berger, our papers, and the classic
paper of Benjamini and Hochberg (1995). The models involve sparse mixtures for
a huge number of parameters and independent tests with a relatively simple
structure . I will briefly discuss the Full Bayes, Empirical Bayes and the
classical BH rule, and their similarities.
In the second part of the talk I
will introduce a Bayesian Oracle providing a lower bound for
misclassification probability and show that the BH rule attains the lower
bound asymptotically for almost a full spectrum of sparsity. Such results are
still out of bounds for Full Bayes and Empirical Bayes.
These results clarify why (and when), possibly contrary to intution, the BH rule is attractive from a decision theoretic point of view.
This is joint work with Malgorzata Bogdan and Arijit Chakrabarti.