One-Shot Learning of Poisson Distributions in Serial Analysis of Gene Expression

Abstract

Traditionally, studies in learning theory tend to concentrate on situations where potentially ever increasing number of training examples is available. However, there are situations where only extremely small samples can be used in order to perform an inference. In such situations it is of utmost importance to theoretically analyze what and under what circumstances can be learned. One such scenario is detection of differentially expressed genes. In our previous study (BMC Bioinformatics, 2009) we theoretically analyzed one of the most popular techniques for identifying genes with statistically different expression in SAGE libraries - the Audic-Claverie statistic (Genome Research, 1997). When comparing two libraries in the Audic-Claverie framework, it is assumed that under the null hypothesis their tag counts come from the same underlying (unknown) Poisson distribution. Since each SAGE library represents a single measurement, the inference has to be performed on the smallest sample possible - sample of size 1. In this contribution we compare the Audic-Claverie approach with a (regularized) maximum likelihood (ML) framework. We analytically approximate the expected K-L divergence from the true unknown Poisson distribution to the model and show that while the expected K-L divergence to the ML-estimated models seems to be always larger than that of the Audic-Claverie statistic, the most divergence appears for true Poisson distributions with small mean parameter. We also theoretically analyze the effect of regularization of ML estimates in the case of zero observed counts. Our results constitute a rigorous analysis of a situation of great practical importance where the benefits of Bayesian approach can be clearly demonstrated in a quantitative and principled manner.