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LI MA - Duke University

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A nonparametric process for multi-resolution adaptive shrinkage
31 October 2013 from 4:00 PM to 5:00 PM
201 Thomas Bldg.
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We introduce an extension to the Polya tree and optional Polya tree processes for generating random probability distributions. By incorporating latent shrinkage states and the transition among these states on different parts of the sample space in a Markovian fashion, inference under this process achieves resolution and location specific adaptive shrinkage. The inference is resolution-specific in the sense that the prior applies a varying amount of shrinkage to data structures of different scales. It is also location specific in that the appropriate amount of shrinkage is determined through the local behavior of the data and so structures of the same scale may receive different amounts of shrinkage. We study the theoretical properties of the process. In particular, we show that it possesses large support, posterior conjugacy, and posterior consistency. We provide analytic recipes for marginalization and for computing the posterior through recursion. We also show that the process can be centered around any parametric family, and is invariant under change-of-variable. We use the last two properties to derive a convenient strategy for inference using mixtures of the process. We present a finite version of the process based on truncating the partition of the sample space at a given resolution, and establish its posterior consistency. We illustrate how to achieve multi-resolution adaptive shrinkage using the process through two numerical examples in density estimation. I will briefly talk about further extensions to multi-sample comparison problems, and computational strategies for scaling to high dimensional situations as well.