In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and show that the asymptotic distance is (d / 2 X l o g n) + c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D,,/n converges to zero at rate (l o g n) / n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stock-market portfolio selection.
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Motivation: Global expression patterns within cells are used for purposes ranging from the identification of disease biomarkers to basic understanding of cellular processes. Unfortunately, tissue samples used in cancer studies are usually composed of multiple cell types and the non-cancerous portions can significantly affect expression profiles. This severely limits the conclusions that can be made about the specificity of gene expression in the cell-type of interest. However, statistical analysis can be used to identify differentially expressed genes that are related to the biological question being studied.Results: We propose a statistical approach to expression deconvolution from mixed tissue samples in which the proportion of each component cell type is unknown. Our method estimates the proportion of each component in a mixed tissue sample; this estimate can be used to provide estimates of gene expression from each component. We demonstrate our technique on xenograft samples from breast cancer research and publicly available experimental datasets found in the National Center for Biotechnology Information Gene Expression Omnibus repository.Availability: R code (http://www.r-project.org/) for estimating sample proportions is freely available to non-commercial users and available at http://www.med.miami.edu/medicine/x2691.xmlContact: jclarke@med.miami.edu
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