We consider sparse Bayesian estimation in the classical multivariate linear regression model with p regressors and q response variables. In univariate Bayesian linear regression with a single response y, shrinkage priors which can be expressed as scale mixtures of normal densities are popular for obtaining sparse estimates of the coefficients. In this paper, we extend the use of these priors to the multivariate case to estimate a p × q coefficients matrix B. We derive sufficient conditions for posterior consistency under the Bayesian multivariate linear regression framework and prove that our method achieves posterior consistency even when p > n and even when p grows at nearly exponential rate with the sample size. We derive an efficient Gibbs sampling algorithm and provide the implementation in a comprehensive R package called MBSP. Finally, we demonstrate through simulations and data analysis that our model has excellent finite sample performance.

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Nonparametric varying coefficient (NVC) models are widely used for modeling time-varying effects on responses that are measured repeatedly. In this paper, we introduce the nonparametric varying coefficient spike-and-slab lasso (NVC-SSL) for Bayesian estimation and variable selection in NVC models. The NVC-SSL simultaneously estimates the functionals of the significant time-varying covariates while thresholding out insignificant ones. Our model can be implemented using a highly efficient expectation-maximization (EM) algorithm, thus avoiding the computational burden of Markov chain Monte Carlo (MCMC) in high dimensions. In contrast to frequentist NVC models, hardly anything is known about the large-sample properties for Bayesian NVC models. In this paper, we take a step towards addressing this longstanding gap between methodology and theory by deriving posterior contraction rates under the NVC-SSL model when the number of covariates grows at nearly exponential rate with sample size. Finally, we illustrate our methodology through simulation studies and data analysis.

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The linear varying coefficient (VC) model generalizes the conventional linear model by allowing the additive effect of each covariate on the outcome to vary as a function of additional effect modifiers. While there are many existing procedures for VC modeling with a single scalar effect modifier (often assumed to be time), there has, until recently, been comparatively less development for settings with multivariate modifiers. Unfortunately, existing state-of-the-art procedures that can accommodate multivariate modifiers typically make restrictive structural assumptions about the covariate effect functions or require intensive problem-specific hand-tuning that scales poorly to large datasets. In response, we propose VC-BART, which estimates the covariate effect functions in a VC model using Bayesian Additive Regression Trees (BART).On several synthetic and real-world data sets, we demonstrate that, with simple default hyperparameter settings, VC-BART displays covariate effect recovery performance superior to state-of-the-art VC modeling techniques and predictive performance on par with more flexible but less interpretable nonparametric regression procedures. We further demonstrate the theoretical near-optimality of VC-BART by synthesizing recent theoretical results about the VC model and BART to derive posterior concentration rates in settings with independent and correlated errors. An R package implementing VC-BART is available at https://github.com/skdeshpande91/VCBART

We revisit the problem of simultaneously testing the means of n independent normal observations under sparsity. We take a Bayesian approach to this problem by studying a scale-mixture prior known as the normal-beta prime (NBP) prior. To detect signals, we propose a hypothesis test based on thresholding the posterior shrinkage weight under the NBP prior. Taking the loss function to be the expected number of misclassified tests, we show that our test procedure asymptotically attains the optimal Bayes risk when the signal proportion p is known. When p is unknown, we introduce an empirical Bayes variant of our test which also asymptotically attains the Bayes Oracle risk in the entire range of sparsity parameters p ∝ n − , ∈ (0, 1). Finally, we also consider restricted marginal maximum likelihood (REML) and hierarchical Bayes approaches for estimating a key hyperparameter in the NBP prior and examine multiple testing under these frameworks.

OBJECTIVE: To investigate the association between individual-level and neighborhood-level risk factors and severe maternal morbidity. METHODS: This was a retrospective cohort study of all pregnancies delivered between 2010 and 2017 in the University of Pennsylvania Health System. International Classification of Diseases codes classified severe maternal morbidity according to the Centers for Disease Control and Prevention guidelines. Logistic regression modeling evaluated individual-level risk factors for severe maternal morbidity, such as maternal age and preeclampsia diagnosis. Additionally, we used spatial autoregressive modeling to assess Census-tract, neighborhood-level risk factors for severe maternal morbidity such as violent crime and poverty. RESULTS: Overall, 63,334 pregnancies were included, with a severe maternal morbidity rate of 2.73%, or 272 deliveries with severe maternal morbidity per 10,000 delivery hospitalizations. In our multivariable model assessing individual-level risk factors for severe maternal morbidity, the magnitude of risk was highest for patients with a cesarean delivery (adjusted odds ratio [aOR] 3.50, 95% CI 3.15–3.89), stillbirth (aOR 4.60, 95% CI 3.31–6.24), and preeclampsia diagnosis (aOR 2.71, 95% CI 2.41–3.03). Identifying as White was associated with lower odds of severe maternal morbidity at delivery (aOR 0.73, 95% CI 0.61–0.87). In our final multivariable model assessing neighborhood-level risk factors for severe maternal morbidity, the rate of severe maternal morbidity increased by 2.4% (95% CI 0.37–4.4%) with every 10% increase in the percentage of individuals in a Census tract who identified as Black or African American when accounting for the number of violent crimes and percentage of people identifying as White. CONCLUSION: Both individual-level and neighborhood-level risk factors were associated with severe maternal morbidity. These factors may contribute to rising severe maternal morbidity rates in the United States. Better characterization of risk factors for severe maternal morbidity is imperative for the design of clinical and public health interventions seeking to lower rates of severe maternal morbidity and maternal mortality.