Proof of Proposition 2.3. Let N td denote the number of nodes at depth d in tree structure S t . Then |S t | = ∞ d=0 N td , and by monotone convergence we must show that E(N td ) is summable. But

Summary Ensembles of decision trees are a useful tool for obtaining flexible estimates of regression functions. Examples of these methods include gradient‐boosted decision trees, random forests and Bayesian classification and regression trees. Two potential shortcomings of tree ensembles are their lack of smoothness and their vulnerability to the curse of dimensionality. We show that these issues can be overcome by instead considering sparsity inducing soft decision trees in which the decisions are treated as probabilistic. We implement this in the context of the Bayesian additive regression trees framework and illustrate its promising performance through testing on benchmark data sets. We provide strong theoretical support for our methodology by showing that the posterior distribution concentrates at the minimax rate (up to a logarithmic factor) for sparse functions and functions with additive structures in the high dimensional regime where the dimensionality of the covariate space is allowed to grow nearly exponentially in the sample size. Our method also adapts to the unknown smoothness and sparsity levels, and can be implemented by making minimal modifications to existing Bayesian additive regression tree algorithms.

Bayesian additive regression trees (BART) provides a flexible approach to fitting a variety of regression models while avoiding strong parametric assumptions. The sum-of-trees model is embedded in a Bayesian inferential framework to support uncertainty quantification and provide a principled approach to regularization through prior specification. This article presents the basic approach and discusses further development of the original algorithm that supports a variety of data structures and assumptions. We describe augmentations of the prior specification to accommodate higher dimensional data and smoother functions. Recent theoretical developments provide justifications for the performance observed in simulations and other settings. Use of BART in causal inference provides an additional avenue for extensions and applications. We discuss software options as well as challenges and future directions.

Background: Besides its intrinsic value as a fundamental nuclear-structure observable, the weak-charge density of 208 Pb-a quantity that is closely related to its neutron distribution-is of fundamental importance in constraining the equation of state of neutron-rich matter.Purpose: To assess the impact that a second electroweak measurement of the weak-charge form factor of 208 Pb may have on the determination of its overall weak-charge density.Methods: Using the two putative experimental values of the form factor, together with a simple implementation of Bayes' theorem, we calibrate a theoretically sound-yet surprisingly little known-symmetrized Fermi function, that is characterized by a density and form factor that are both known exactly in closed form.Results: Using the charge form factor of 208 Pb as a proxy for its weak-charge form factor, we demonstrate that using only two experimental points to calibrate the symmetrized Fermi function is sufficient to accurately reproduce the experimental charge form factor over a significant range of momentum transfers. Conclusions:It is demonstrated that a second measurement of the weak-charge form factor of 208 Pb supplemented by a robust theoretical input in the form of the symmetrized Fermi function, would place significant constraints on the neutron distribution of 208 Pb. In turn, such constraints will become vital in the interpretation of hadronic experiments that will probe the neutron-rich skin of exotic nuclei at future radioactive beam facilities.

We develop a Bayesian nonparametric model for a longitudinal response in the presence of nonignorable missing data. Our general approach is to first specify a working model that flexibly models the missingness and full outcome processes jointly. We specify a Dirichlet process mixture of missing at random (MAR) models as a prior on the joint distribution of the working model. This aspect of the model governs the fit of the observed data by modeling the observed data distribution as the marginalization over the missing data in the working model. We then separately specify the conditional distribution of the missing data given the observed data and dropout. This approach allows us to identify the distribution of the missing data using identifying restrictions as a starting point. We propose a framework for introducing sensitivity parameters, allowing us to vary the untestable assumptions about the missing data mechanism smoothly. Informative priors on the space of missing data assumptions can be specified to combine inferences under many different assumptions into a final inference and accurately characterize uncertainty. These methods are motivated by, and applied to, data from a clinical trial assessing the efficacy of a new treatment for acute Schizophrenia.

Missing data is almost always present in real datasets, and introduces several statistical issues. One fundamental issue is that, in the absence of strong uncheckable assumptions, effects of interest are typically not nonparametrically identified. In this article, we review the generic approach of the use of identifying restrictions from a likelihood-based perspective, and provide points of contact for several recently proposed methods. An emphasis of this review is on restrictions for nonmonotone missingness, a subject that has been treated sparingly in the literature. We also present a general, fully-Bayesian, approach which is widely applicable and capable of handling a variety of identifying restrictions in a uniform manner.

In longitudinal clinical trials, one often encounters missingness which is thought to be non-ignorable. It is well-known that non-ignorable missingness introduces fundamental identifiability issues, resulting in intention-to-treat effects being unidentified; the best one can do is to conduct a sensitivity analysis to assess how much of the inference is being driven by missingness. We introduce a Bayesian nonparametric framework for conducting inference in the presence of non-ignorable, non-monotone missingness. This framework focuses on the specification of an auxiliary prior, which we refer to as a working prior, on the space of complete data generating mechanisms. This prior is not used to conduct inference, but instead is used to induce a prior on the observed data generating mechanism, which is then used in conjunction with an identifying restriction to conduct inference. Advantages of this approach include a flexible modeling framework, access to simple computational methods, strong theoretical support, straightforward sensitivity analysis, and its applicability to non-monotone missingness.

Tree-based regression and classification ensembles form a standard part of the data-science toolkit. Many commonly used methods take an algorithmic view, proposing greedy methods for constructing decision trees; examples include the classification and regression trees algorithm, boosted decision trees, and random forests. Recent history has seen a surge of interest in Bayesian techniques for constructing decision tree ensembles, with these methods frequently outperforming their algorithmic counterparts. The goal of this article is to survey the landscape surrounding Bayesian decision tree methods, and to discuss recent modeling and computational developments. We provide connections between Bayesian tree-based methods and existing machine learning techniques, and outline several recent theoretical developments establishing frequentist consistency and rates of convergence for the posterior distribution. The methodology we present is applicable for a wide variety of statistical tasks including regression, classification, modeling of count data, and many others. We illustrate the methodology on both simulated and real datasets.