Subhashis Ghoshal is a professor of statistics at North Carolina State University, Raleigh. His research interests span many areas including Bayesian statistics, asymptotics, nonparametrics and high dimensional models, with diverse applications. In particular, his pioneering work on concentration of posterior distributions led to theoretical understanding of nonparametric Bayesian procedures. He was honored with fellowship from the IMS (2006), ASA (2010) and ISBA (2016). He has received several awards, including the P.C. Mahalanobis Gold Medal (1990), Indian Science Congress Young Scientist Award (1995), NSF Career Award (2003), Sigma-Xi Research Award (2004), International Indian Statistical Association Young Researcher Award (2007) and Cavell Brownie Mentoring Award (2015). He held the honorary positions of Eurandom Chair (2010–11) and the Royal Netherlands Academy Arts and Sciences Visiting Professorship (2013–14). His research has been supported by several US federal funding agencies, European granting institutions and industry grants. He serves or has served on the editorial boards of many leading statistics journals including the Annals of Statistics, Bernoulli, Electronic Journal of Statistics and Sankhya. Seventeen doctoral students so far have graduated under his advising.
Subhashis Ghoshal’s Medallion lecture will be given at the 2017 Joint Statistical Meetings in Baltimore (July 29–August 4, 2017). See the online program at http://ww2.amstat.org/meetings/jsm/2017/onlineprogram/index.cfm
Coverage of Nonparametric Credible Sets
Subhashis Ghoshal’s Medallion lecture will discuss frequentist coverage properties of Bayesian credible sets for nonparametric and high dimensional models. Bayesians and frequentists quantify uncertainty in very different ways. While a Bayesian’s uncertainty quantification is based on a direct probability assessment of the parameters and thus has excellent interpretation in a conditional framework, potential strong dependence on the prior may lead to confusion. This necessitates the study of frequentist coverage of Bayesian credible sets. In the classical setting of repeated sampling from fixed dimensional smooth parametric families, the celebrated Bernstein–von Mises theorem asserts that the posterior distribution of the parameter is asymptotically normal with mean at the maximum likelihood estimator and variance the inverse of Fisher information. The most important consequence of this result is that the coverage of a Bayesian credible set approximately matches its credibility, and hence Bayesians and frequentists are in agreement about uncertainty quantification. Such a matching continues in many other fixed dimensional parametric families which are not smoothly parameterized as well as in models where the number of parameters grows to infinity sufficiently slowly. In certain nonparametric problems with parametric convergence rate (like that of estimating a distribution function), empirical estimators typically have Gaussian process limits by Donkser-type theorems. For certain priors, the posterior of the function centered at the empirical estimator can have the same limit, thus again Bayesian credible sets will have asymptotically valid coverage. A similar behavior is observed in many semi-parametric models for the parametric part, or for certain differentiable functionals of the function indexing a nonparametric model. In certain parametric models described by structural relations like in differential equation models, a type of “projection posterior” distributions gives credible sets with valid asymptotic frequentist coverage. However, in smoothing problems with interest in the whole function, all these niceties go away, even for the simple signal plus white noise model although a conjugate prior is immediately available. Under optimal smoothing to produce the best convergence rate, posterior credible regions may have arbitrarily low asymptotic frequentist coverage. The reason for such an anomaly is that the order of the bias under optimal smoothing matches the order of posterior variability, thus poorly centering posterior credible sets and spoiling coverage. Interestingly, the disagreement between Bayesian credibility and frequentist coverage at fixed credibility level may go away in the high credibility regime. In general the problem may be resolved by slightly inflating Bayesian credible sets, especially if uniform credible bands are desired. Assuring frequentist coverage of a Bayesian credible set which adapts its size to the smoothness of the underlying true function is a lot more subtle. The only possible way to maintain both coverage and size is to discard certain “deceptive parameters” from consideration which lead to “excessive bias” in the procedure. Then optimal sized inflated credible sets with guaranteed asymptotic coverage can be obtained. The talk will be concluded by considering a setup of shape-restricted models, for which it is observed that asymptotic coverage of a Bayesian credible set can be obtained explicitly but it may differ from the corresponding frequentist coverage. A simple modification of a credible set is devised to guarantee a desired coverage level.