SBSS Student Paper Award I

Model misspecification is problematic for Bayesians. Various methods have been proposed to modify the update from prior distribution to posterior distribution when one acknowledges that one's model is imperfect. Two current proposals are Bayesian restricted likelihood (BRL) methods and generalized Bayesian (GB) methods. The first focuses on aspects of the model that are believed to be modeled well and derives the posterior distribution by conditioning on an insufficient statistic (e.g., Huber's M-estimate) that captures those aspects of the model. The second focuses on a particular inference, making use of a loss function to define the target of inference. The usual Bayesian update is altered: the likelihood function is replaced with the exponentiated negative loss. We compare these two approaches by investigating both finite sample and asymptotic behavior when the data come from a location family. Suggestions for a choice between these two methods are given in different cases. 

Variational Bayes (VB) is a popular and computationally efficient method to approximate the posterior distribution in Bayesian inference, especially when the exact posterior is analytically intractable and sampling-based approaches are computationally prohibitive. While VB often yields accurate point estimates, its uncertainty quantification (UQ) is known to be unreliable. For example, credible intervals derived from VB posteriors tend to exhibit undercoverage, failing to achieve nominal frequentist coverage probabilities. In this article, we address this challenge by proposing Trustworthy Variational Bayes (TVB), a method to recalibrate the UQ of broad classes of VB procedures. Our approach follows a bend-to-mend strategy: we intentionally misspecify the likelihood (bend) to correct VB's flawed UQ (mend). In particular, we first relax VB by building on a recently proposed fractional VB method indexed by a fraction ω, and then identify the optimal fraction parameter using conformal techniques such as sample splitting and bootstrapping. This yields recalibrated UQ for any given parameter of interest. On the theoretical side, we establish that the calibrated credible intervals achieve asymptotically correct frequentist coverage for a given parameter of interest; this, to the best of our knowledge, is the first such theoretical guarantee for VB. On the practical side, we introduce the "TVB table", which enables (1) massive parallelization and remains agnostic to the parameter of interest during its construction, and (2) efficient post-hoc identification of the optimal fraction parameter for any specified parameter of interest. The proposed method is illustrated via Gaussian mixture models and Bayesian mixture linear regression models, and numerical experiments demonstrate that TVB method outperforms standard VB and achieves normal frequentist coverage in finite samples. 

Dirichlet distributions are commonly used for modeling vectors in a probability simplex. When used as a prior or a proposal distribution, it is natural to set the mean of a Dirichlet to be equal to the location where one wants the distribution to be centered. However, if the mean is near the boundary of the probability simplex, then a Dirichlet distribution becomes highly concentrated either (i) at the mean or (ii) extremely close to the boundary. Consequently, centering at the mean provides poor control over the location and scale near the boundary. In this article, we introduce a method for improved control over the location and scale of Beta and Dirichlet distributions. Specifically, given a target location point and a desired scale, we maximize the density at the target location point while constraining a specified measure of scale. We consider various choices of scale constraint, such as fixing the concentration parameter, the mean cosine error, or the variance in the Beta case. In several examples, we show that this maximum density method provides superior performance for constructing priors, defining Metropolis-Hastings proposals, and generating simulated probability vectors.  

Joint species distribution models are popular in ecology for modeling covariate effects on species occurrence, while characterizing cross-species dependence. Data consist of multivariate binary indicators of the occurrences of different species in each sample, along with sample-specific covariates. A key problem is that current models implicitly assume that the list of species under consideration is predefined and finite, while for highly diverse groups of organisms, it is impossible to anticipate which species will be observed in a study and discovery of unknown species is common. This article proposes a new modeling paradigm for statistical ecology, which generalizes traditional multivariate probit models to accommodate large numbers of rare species and new species discovery. We discuss theoretical properties of the proposed modeling paradigm and implement efficient algorithms for posterior computation. Simulation studies and applications to fungal biodiversity data provide compelling support for the new modeling class. 

Although it is an extremely effective, easy-to-use, and increasingly popular tool for nonparametric regression, the Bayesian Additive Regression Trees (BART) model is limited by the fact that it can only produce discontinuous output. Initial attempts to overcome this limitation were based on regression trees that output Gaussian Processes instead of constants. Unfortunately, implementations of these extensions cannot scale to large datasets. We propose ridgeBART, an extension of BART built with trees that output linear combinations of ridge functions (i.e., a composition of an affine transformation of the inputs and non-linearity); that is, we build a Bayesian ensemble of localized neural networks with a single hidden layer. We develop a new MCMC sampler that updates trees in linear time and establish nearly minimax-optimal posterior contraction rates when the underlying function is piecewise smooth. We demonstrate ridgeBART's effectiveness on synthetic data and use it to estimate the probability that a professional basketball player makes a shot from any location on the court in a spatially smooth fashion.