|Stochastic Systems Group|
Prof. Eric D. Kolaczyk
Department of Mathematics and Statistics, Boston University
Wavelets, recursive partitioning, and graphical models represent three frameworks for modeling signals and images that have proven to be highly successful in a variety of application areas. Each of these frameworks has certain strengths and weaknesses, often complementary among them. In this talk I will present a framework for a certain class of multiscale probability models that simultaneously shares characteristics with all three of these frameworks. These models are grounded upon a common factorized form for the data likelihood into a product of components with localized information in position and scale -- similar to a wavelet decomposition. In fact, such factorizations can be linked formally with a probabilistic analogue of a multiresolution analysis. Efficient algorithms for estimation and segmentation can be built upon these factorizations, with direct parallels to thresholding, partitioning, and probability propagation algorithms in the existing literature. Finally, estimators deriving from this framework can be shown to have near-optimality and adaptivity properties similar to those now-classical results established for wavelets in the Gaussian signal-plus-noise model, but for other distributions as well, such as Poisson and multinomial. As time allows, I will illustrate the application of our framework using a handful of examples from high-energy astrophysics and census geography.
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