Jayanta K. Ghosh Memorial Session on Bayesian Nonparametrics, Empirical Processes, and Convexity

Organizer: James O. Berger, The Arts and Sciences Professor of Statistics, Department of Statistical Science, Duke University

Chair: R.V. Ramamoorthi, Professor of Statistics and Probability, Department of Statistics and Probability, Michigan State University

Speakers

Anirban DasGupta, Professor of Statistics, Department of Statistics, Purdue University
Subhashis Ghosal, Professor, Department of Statistics, North Carolina State University
Surya Tokdar, Associate Professor, Department of Statistical Science, Duke University

Schedule

Friday, June 8, 1:30-3:30 p.m. in STEW 214 AB

Time	Speaker	Title
1:30-2:10 p.m.	Anirban DasGupta	Empirical Processes, Clustering, Convexity, and Variational Diameters
	Abstract: This talk is tenderly dedicated to the many memories of the speaker of his 44 years of relationship with his PhD thesis advisor Professor Jayanta K. Ghosh during 1980-83 at the Indian Statistical Institute in Calcutta. He very gratefully acknowledges the education and insights he received from Professor Ghosh. The talk is split into two unrelated sets of results. In the first half of the talk, the speaker will formulate the so called three-sigma rule of elementary statistics texts in the language of modern empirical process theory and will proceed to define a cluster functional for one dimensional iid data. He will address weak convergence of the cluster functional and connect it to the stable domain of attraction of the underlying CDF. He will show weak convergence to a Gaussian process under enough moments, and give the covariance kernel in explicit form, different from that of the usual Brownian Bridge. He will propose the supremum of the unsigned cluster functional as a test statistic, and he will address tail probabilities of itby using covering numbers of the time set, the standard modern technique for handling tails of suprema of Gaussian processes. He will leave an open problem for the audience to solve. In the second half of the talk, he will connect variational inequalities to variational diameters of sets of probability distributions. An example of such a problem is to bound the diameter of a class of densities on a compact or unbounded interval on the line if the densities all possess certain monotonicity, or other shape properties. He will use analytic inequalities on norms of such functions or norms of derivatives of such functions to variational distance between two such functions. There is a large family of such analytic inequalities, all classic, e.g., inequalities of Kolmogorov-Landau to those of Ostrowski, Brascamp-Lieb and Beckner, but lying mostly unused in statistics. He will recall that variational (via Kullback-Leibler) distances are classically related to minimax rates, power of tests, and local limit theorems, which are a few reasons for a statistician to consider variational inequalities.
2:10-2:50 p.m.	Subhashis Ghosal	Bayesian mode and maximum estimation and accelerated rates of contraction
	Abstract: We study the problem of estimating the mode and maximum of an unknown regression function in the presence of noise. We adopt the Bayesian approach by using tensor product B-splines and endowing the coefficients with Gaussian priors. In the usual fixed-in-advanced sampling plan, we establish posterior contraction rates for mode and maximum and show that they coincide with the minimax rates for this problem. To quantify estimation uncertainty, we construct credible sets for these two quantities that have high coverage probabilities with optimal sizes. If one is allowed to collect data sequentially, we further propose a Bayesian two-stage estimation procedure, where a second stage posterior is built based on samples collected within a credible set arising from a first stage posterior. Under appropriate conditions on the radius of this credible set, we can accelerate optimal contraction rates in the fixed-in-advanced setting to the minimax sequential rates. A simulation experiment shows that our Bayesian two-stage procedure outperforms single-stage procedure and also slightly improves upon a non-Bayesian two-stage procedure, due to shrinkage effect of our Bayesian estimators.
2:50-3:30 p.m.	Surya Tokdar	Semi-parametric density estimation with logistic Gaussian processes
	Abstract: During late 70s to early 90s, Tom Leonard and Peter Lenk introduced the logistic Gaussian process prior for nonparametric Bayesian density estimation. In this framework, one treats the logarithm of the unknown density function as the model parameter. The log-density is assumed to belong to the space of continuous functions, over which the law of a smooth Gaussian process is taken to give the prior measure. In the 2000s, I worked with Professor JK Ghosh to operationalize this program and to study the frequentist asymptotic properties of the resulting estimation method. The method was initially developed as a purely nonparametric estimation technique suited for estimating an unknown density over a known compact set, such as the unit interval. Later, we introduced and studied a semi-parametric extension that was suitable for estimating densities with unknown, and, possibly unbounded support. The main innovation involved using a parametric family of CDFs to transform the unbounded data to the unit interval and then applying the old purely nonparametric model on the pdf of the transformed data. The Bayesian framework allows joint estimation where both the transformation parameter, and, the density on the unit interval are estimated simultaneously. An important by-product is that the tail index of the semi-parametric density exactly matches the tail index of the CDF used in the transformation. In my recent work, I have studied the application of this model in estimating heavy tailed densities, and, forecasting rare events such as the 1000 year flood based on limited precipitation data. I will present some new theoretical results concerning asymptotic posterior contraction rates for the semiparametric model, as well as, recovery of the tails.