Seminars

Bayesian modeling is ubiquitous in large-scale problems even when frequentist criteria are in mind for evaluating the performance of a procedure. In particular, regularized estimation methods, that may be derived by eliciting a prior distribution on the model parameters, have been shown especially effective for analyzing large data. Appealing to Robbins’s compound decision theory, we introduce a theoretical framework for deriving optimal Bayes rules in which the prior distribution consists of permutations of the parameter vector. For the special case of “symmetric” statistical problems, we show that our Bayes rules also minimize the frequentist Risk for any fixed parameter vector configuration. Our main applicable contribution is the introduction of nonparametric deconvolution methodology, based on hierarchical Bayes modeling, that approximates the marginal parameter distribution. We use this methodology to approximate the theoretical Bayes rule. Our methodology is shown to be particularly effective in low-signal high-dimensional problems in which, even though it is difficult to estimate the components of the parameter vector, we are still able to tease out the marginal distribution of the parameter vector and thus, the resulting Bayes rules perform better than state of the art shrinkage estimators. Furthermore, as large-scale problems tend to be approximately symmetric, our Bayes rules provide near-optimal frequentist performance. For concreteness and clarity, I will present the theoretical framework and hierarchical Bayes modeling for a High-dimensional logistic regression and demonstrate its application on several simulated examples.

The general motivation standing behind this research is to understand relationships between dynamical and model-theoretic properties of definable [topological] groups and between dynamical properties of groups of automorphisms of first order structures and model-theoretic properties of the underlying theories. More specifically, our goal is to understand model-theoretic consequences of various notions of amenability.

Among the notions of amenability that we are interested in are: definable amenability of a definable group, classical amenability of a topological group, and, more generally, [weak] definable topological amenability of a definable topological group. We also introduce and study amenable theories.

The consequences of amenability that we obtain are the appropriate versions of G-compactness: for first order theories this is the equality of Lascar strong types and Kim-Pillay strong types; for definable [topological] groups this is the equality of suitably defined connected components $G^{000}$ and $G^{00}$ of the group $G$ in question.

Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions.

My series of talks will be based on my preprint “Amenability and definability” joint with Ehud Hrushovski and Anand Pillay. In the first series of talks, I will focus on the context of definable [topological] groups; the second series will be devoted to our new notion of amenable theory.

Let us recall that a compact space K is Eberlein compact if it can be embedded into some Banach space X equipped with the weak topology. Our talk will be devoted to the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. Several such spaces were obtained using some additional set-theoretic assumptions. Recently, P. Koszmider constructed the first such example in ZFC. We investigate this problem for the class of Eberlein compact spaces. We construct such Eberlein compacta, assuming the existence of a Luzin set. We also show that it is consistent with ZFC that each Eberlein compact space of weight greater than $\omega_1$ contains a nonmetrizable closed zero-dimensional subspace. The talk is based on the paper "On two problems concerning Eberlein compacta":