Ray Bai

The linear varying coefficient models posits a linear relationship between an outcome and covariates in which the covariate effects are modeled as functions of additional effect modifiers. Despite a long history of study and use in statistics and econometrics, state-of-the-art varying coefficient modeling methods cannot accommodate multivariate effect modifiers without imposing restrictive functional form assumptions or involving computationally intensive hyperparameter tuning. In response, we introduce VCBART, which flexibly estimates the covariate effect in a varying coefficient model using Bayesian Additive Regression Trees. With simple default settings, VCBART outperforms existing varying coefficient methods in terms of covariate effect estimation, uncertainty quantification, and outcome prediction. We illustrate the utility of VCBART with two case studies: one examining how the association between later-life cognition and measures of socioeconomic position vary with respect to age and socio-demographics and another estimating how temporal trends in urban crime vary at the neighborhood level. An R package implementing VCBART is available at https://github.com/skdeshpande91/VCBART.

Sparse linear regression methods for high-dimensional data commonly assume that errors have constant variance, which can be violated in practice. For example, Aphasia Quotient (AQ) is a critical measure of language impairment and informs treatment decisions, but it is challenging to measure in stroke patients. It is of interest to use high-resolution T2 neuroimages of brain damage to predict AQ. However, sparse regression models show marked evidence of heteroscedastic error even after transformations are applied. This violation of the homoscedasticity assumption can lead to biased and inconsistent standard errors of estimated coefficients and prediction intervals (PI) with improper length. Bayesian heteroscedastic linear regression models relax the homoscedastic error assumption but can enforce restrictive prior assumptions on parameters, and many are computationally infeasible in the high-dimensional setting. This paper proposes estimating high-dimensional heteroscedastic linear regression models using a heteroscedastic partitioned empirical Bayes Expectation Conditional Maximization (H-PROBE) algorithm. H-PROBE is a computationally efficient maximum a posteriori estimation approach that requires minimal prior assumptions and can incorporate covariates known or hypothesized to impact heterogeneity. We apply this method by using high-dimensional neuroimages to predict and provide PIs for AQ that accurately quantify predictive uncertainty. Our analysis demonstrates that H-PROBE can provide narrower PI widths than standard methods without sacrificing coverage. Narrower PIs are clinically important for determining the risk of moderate to severe aphasia. Additionally, through extensive simulation studies, we exhibit that H-PROBE results in superior prediction, variable selection, and predictive inference compared to alternative methods.

Fits linear varying coefficient (VC) models, which assert a linear relationship between an outcome and several covariates but allow that relationship (i.e., the coefficients or slopes in the linear regression) to change as functions of additional variables known as effect modifiers, by approximating the coefficient functions with Bayesian Additive Regression Trees. Implements a Metropolis-within-Gibbs sampler to simulate draws from the posterior over coefficient function evaluations. VC models with independent observations or repeated observations can be fit. For more details see Deshpande et al. (2024) .

Periodontal pocket depth is a widely used biomarker for diagnosing risk of periodontal disease. However, pocket depth typically exhibits skewness and heavy-tailedness, and its relationship with clinical risk factors is often nonlinear. Motivated by periodontal studies, this paper develops a robust single-index modal regression framework for analyzing skewed and heavy-tailed data. Our method has the following novel features: (1) a flexible two-piece scale Student-$t$ error distribution that generalizes both normal and two-piece scale normal distributions; (2) a deep neural network with guaranteed monotonicity constraints to estimate the unknown single-index function; and (3) theoretical guarantees, including model identifiability and a universal approximation theorem. Our single-index model combines the flexibility of neural networks and the two-piece scale Student-$t$ distribution, delivering robust mode-based estimation that is resistant to outliers, while retaining clinical interpretability through parametric index coefficients. We demonstrate the performance of our method through simulation studies and an application to periodontal disease data from the HealthPartners Institute of Minnesota. The proposed methodology is implemented in the \textsf{R} package \href{https://doi.org/10.32614/CRAN.package.DNNSIM}{\textsc{DNNSIM}}.

We introduce a Bayesian framework for mixed-type multivariate regression using continuous shrinkage priors. Our framework enables joint analysis of mixed continuous and discrete outcomes and facilitates variable selection from the p covariates. Theoretical studies of Bayesian mixed-type multivariate response models have not been conducted previously and require more intricate arguments than the corresponding theory for univariate response models due to the correlations between the responses. In this paper, we investigate necessary and sufficient conditions for posterior contraction of our method when p grows faster than sample size n. The existing literature on Bayesian high-dimensional asymptotics has focused only on cases where p grows subexponentially with n. In contrast, we study the asymptotic regime where p is allowed to grow exponentially in terms of n. We develop a novel two-step approach for variable selection which possesses the sure screening property and provably achieves posterior contraction even under exponential growth of p. We demonstrate the utility of our method through simulation studies and applications to real data, including a cancer genomics dataset where n=174 and p=9183. The R code to implement our method is available at https://github.com/raybai07/MtMBSP.

Estimating the mixing density of a latent mixture model is an important task in signal processing. Nonparametric maximum likelihood estimation is one popular approach to this problem. If the latent variable distribution is assumed to be continuous, then bootstrapping can be used to approximate it. However, traditional bootstrapping requires repeated evaluations on resampled data and is not scalable. In this letter, we construct a generative process to rapidly produce nonparametric maximum likelihood bootstrap estimates. Our method requires only a single evaluation of a novel two-stage optimization algorithm. Simulations and real data analyses demonstrate that our procedure accurately estimates the mixing density with little computational cost even when there are a hundred thousand observations.