Advances in Nonparametric Bayesian and Empirical Bayesian Theory and Methods

The causal inference literature has increasingly recognized that explicitly targeting treatment effect heterogeneity can lead to improved scientific understanding and policy recommendations. Towards the same ends, studying the causal pathway connecting the treatment to the outcome can be also useful. This paper addresses these problems in the context of causal mediation analysis. We introduce a varying coefficient model based on Bayesian additive regression trees to identify and regularize heterogeneous causal mediation effects; analogously with linear structural equation models, these effects correspond to covariate-dependent products of coefficients. We show that, even on large datasets with few covariates, LSEMs can produce highly unstable estimates of the conditional average direct and indirect effects, while our Bayesian causal mediation forests model produces estimates that are stable. We find that our approach is conservative, with effect estimates shrunk towards homogeneity.'' We examine the salient properties of our method using both data from the Medical Expenditure Panel Survey and empirically-grounded simulated data. Finally, we show how our model can  

Density estimation is a useful statistical tool for sketching variations of data, profiling information content, and making risk optimal decisions. It consequently plays a fundamental role in a wide spectrum of statistical analyses and applications, such as two-sample comparison, data compression, and nonparametric evaluation of disease risk. In this work, we focus on tree based methods for density estimation under the Bayesian framework. First we show that the Bayesian density tree can achieve minimax convergence over the anisotropic Besov class, which implies that the method can adapt to spatially inhomogeneous features of the underlying density function, and can achieve fast convergence as the dimension increases. We also introduce a novel Bayesian model for density forests, and show that over the anisotropic H{\"o}lder space, forests can achieve faster convergence than trees, in an adaptive way. The Bayesian framework naturally endows a stochastic search scheme over the tree or forest space. For both Bayesian density trees and forests, we provide numerical results to illustrate their performance in the high-dimensional case. 

We present a new way to summarize and select mixture models via the hierarchical clustering tree (dendrogram) constructed from an overfitted latent mixing measure. Our proposed method bridges agglomerative hierarchical clustering and mixture modeling. The dendrogram's construction is derived from the theory of convergence of the mixing measures, and as a result, we can both consistently select the true number of mixing components and obtain the pointwise optimal convergence rate for parameter estimation from the tree, even when the model parameters are only weakly identifiable. In theory, it explicates the choice of the optimal number of clusters in hierarchical clustering. In practice, the dendrogram reveals more information on the hierarchy of subpopulations compared to traditional ways of summarizing mixture models. Several simulation studies are carried out to support our theory. We also illustrate the methodology with an application to single-cell RNA sequence analysis. 

We study multiple testing in the normal means problem with estimated variances that are shrunk through empirical Bayes methods. The situation is asymmetric in that a prior is posited for the nuisance
parameters (variances) but not the primary parameters (means). If the prior were known, one could proceed by computing p-values conditional on sample variances; a strategy called partially Bayes inference by Sir David Cox. These conditional p-values satisfy a Tweedie-type formula and are approximated at nearly-parametric rates when the prior is estimated by nonparametric maximum likelihood. If the variances are in fact fixed, the approach retains type-I error guarantees. As is common in the empirical Bayes paradigm, our results hinge on the interpretation of the prior as the frequency distribution of the nuisance parameters, and should be contrasted with e.g., the conditional predictive p-values of Bayarri and Berger. Based on joint work with Bodhisattva Sen.