Bayesian NonParametrics

Speaker(s)

David Dunson (Duke University)
R.V. Ramamoorthi (Michigan State University)
M.J. Bayarri (University of Valencia, Spain, and Duke University)
Peter J. Bickel (University of California, Berkeley)

Description

Bayesian Nonparametrics is passing through a new phase of fast theoretical development as well as novel practical applications to bioinformatics, survival analysis, clustering, document analysis, computer models for complex phenomena, and variable selection in different scenarios.Some of the best people in Bayesian Theory and Applications will be here to speak on areas where they have made major contributions. The Theory That Would Not Die (Sharon Bertsch Mcgrayne, 2011) will be seen at its best.

Schedule

Fri, June 22 - Location: STEW 202

Time	Speaker	Title

8:30-9:15AM	David Dunson	Sparse Nonparametric Bayesian Learning from Big Data
	Abstract: In modern applications, data sets tend to be big and highly structured, with large p, small n problems commonly encountered. In such settings, sparse representations of the data are crucial and there is a rich frequentist literature focused on inducing sparsity through penalization (typically L1). Motivated by genetic epidemiology and imaging applications, we instead develop nonparametric Bayesian methods that avoid parametric assumptions while favoring low-dimensional representations of complex high-dimensional data. In this talk, the particular focus is on Bayesian probabilistic tensor factorizations, which generalize low rank matrix factorizations, such as SVD, to higher orders. The framework accommodates general joint modeling of object data of different types (images, text, categorical, real, etc) but for simplicity we focus on two applications: (1) high-dimensional multivariate categorical data analysis (contingency tables); (2) estimation of lower dimensional manifolds from point cloud data. In the contingency table case, we propose a collapsed Tucker factorization and develop associated methods for testing of associations and interactions in huge sparse tables. In the manifold learning case, we propose a tensor product of basis functions for estimating 3d closed surfaces. In both settings, theoretical results are provided on large support and asymptotic properties & efficient computational methods are developed, which scale to large data sets. (joint work with Anirban Bhattacharya & Debdeep Pati)
9:15-10:00AM	R.V. Ramamoorthi	Posterior consistency of Bayesian quantile regression based on the misspecified asymmetric Laplace density
	Abstract: Asymmetric Lapace distribution is a widely used and empirically verified approach to Bayesian Quantile Regression. In this talk we will explore consistency issues under misspecification that is when the true model is not ALD and the covariates are non-random. We will first look at the simple parametric case and then discuss extensions to nonparametric regression functions. (joint work with Karthik Sriram and Pulak Ghosh both from the Indian Institute of Management, Bangalore, India.)
10:00-10:30AM	Break
10:30-11:15AM	M.J. (Susie) Bayarri	Spatial models in the analysis of data from computer models (UQ)
	Abstract: Computer models, also called "simulators", are becoming increasingly used in virtually all areas of science, engineering, medicine and humanities. Outputs of the computer models are intended to 'mimic' reality, and they are complex, highly non-linear functions of several inputs. Complex computer models are typically very expensive to run, and one single run can take hours or days, rendering them useless for 'brute force' use in sensitivity analyses, optimization, and modern statistical analyses requiring hundreds thousand MCMC runs. The usual statistical approach to analyses involving computer models uses spatial models, fitted with relatively few computer model runs, to 'predict' the output of the simulator at untried inputs. The response surface becomes an "emulator", that is, a very fast surrogate for the simulator, along with an assessment of the error incurred in the approximation. Spatial models are also needed to flexibly model the 'discrepancy term', which is the difference between outputs from the simulator and reality. These and other topics are addressed in this talk, as well as possible extensions and challenges of the spatial modeling.
11:15AM-12:00PM	Peter J. Bickel	The Bernstein von Mises Theorem in non and semiparametric models: Use with caution
	Abstract: The Bernstein von Mises (B vM) Theorem yields asymptotic equivalence of Bayesian and frequentist inference in regular parametric models to order zero(n^-1/2) if the prior has a continuous positive density. Thus, Bayes estimates are locally minimax and efficient while maximum likelihood estimates are universal Bayes. In non and semiparametric models infinite dimensional parameters cannot generally be estimated at rate n^-1/2. Minmax estimation is possible but the minmax rates depend upon the smoothness or other measures of complexity of the distributions permitted under the model. Nevertheless, Ghosh, Ghosal, van der Vaart (2000) and other writers have shown that natural classes of nonparametric Bayes priors can be used to obtain optimal rates. On the other hand, following work of Stein (1956), semiparametric efficiency theory has established that large classes of "smooth" Euclidean valued functions of the infinite dimensional parameters (smooth parameters) can be asymptotically estimated linearly at rate n^-1/2 and, in fact, estimates can be produced which are asymptotically locally minimax and efficient. It was observed in special cases that posterior expectations of some smooth parameters as above were also efficient. Subsequently, Shen (2002) formally and other writers, eg. Ghosal (2000) and others, showed in various examples that this was a consequence of the validity of the B vM theorem for appropriate marginal posterior distributions. In a recent paper, Kleijn and B. (2012) established sharp sufficient conditions under which BvM held in the nonparametric case. In the examples exhibited so far minmaxity of the posterior distribution of the infinity dimensional parameter and B vM for smooth functions of this parameter could be coupled. We will review some of this work but argue by example that achievement of minmaxity of the posterior distribution for infinity dimensional parameters and of BvM for all smooth linear parameters is incompatible. In another example we show that for a well known smooth parameter it would appear that priors which achieve both aims can be constructed but depend on which smooth parameter one wishes to use the posterior for. Finally we give an analogue of Doob's theorem which shows that the prior measure of the set of infinity dimensional parameter values for which n^-1/2 consistency for the marginal posterior of a smooth parameter fails, is zero if there are frequentist ways of n^-1/2 consistent estimation. While this is satisfactory for Bayesians who never doubt their prior the robustness issues we have raised remain. This is joint work with Ya'acov Ritov, A.C. Gamst, and B.J. Kleijn