PhD projects

Minimaxity and Admissibility of Bayesian Neural Networks

Bayesian neural networks (BNNs) offer a natural probabilistic formulation for inference in deep learning models. Despite their popularity, their optimality has received limited attention through the lens of statistical decision theory. In this paper, we study decision rules induced by deep, fully connected feedforward ReLU BNNs in the normal location model under quadratic loss. We show that, for fixed prior scales, the induced Bayes decision rule is not minimax. We then propose a hyperprior on the effective output variance of the BNN prior that yields a superharmonic square-root marginal density, establishing that the resulting decision rule is simultaneously admissible and minimax. We further extend these results from the quadratic loss setting to the predictive density estimation problem with Kullback--Leibler loss. Finally, we validate our theoretical findings numerically through simulation.

You can view the paper here.

Modeling Dynamic Correlation Matrices with Shrinkage Priors

Estimating time-varying correlation matrices is challenging because existing methods may adapt slowly to structural changes, impose insufficient regularization, or produce diffuse posterior uncertainty. In moderate dimensions, an additional difficulty is summarizing the estimated evolving dependence structure for downstream decision-making tasks. We propose a Bayesian approach based on a low-rank factor representation, with latent states evolving under a dynamic shrinkage prior and observation errors following a multivariate factor stochastic volatility model. This specification allows locally adaptive regularization of the estimated correlation structure over time and informative uncertainty quantification. We establish, to our knowledge, a first-of-its-kind posterior contraction result for dynamically regularized Bayesian models, showing contraction around the true model parameters at an explicit rate under averaged Hellinger distance. To summarize the estimated correlation matrices, we build on the information-theoretic concept of total correlation to obtain a scalar measure of cross-sectional dependence. Simulation studies show improved accuracy and responsiveness relative to competing methods in a range of challenging scenarios. We then apply our method to monitoring the correlation evolution of equity portfolios during periods of financial market stress.

You can view the paper here.

Foreclassing: From Forecasts to Decisions

Time series forecasts are widely used to inform decisions under uncertainty. In many applications, these decisions involve classifying a time series based on outcomes that have yet to be observed. In this paper, we introduce Foreclassing, a new machine learning problem in which the class label of a time series is defined as a function of its future observations. We provide a mathematical formulation of Foreclassing and establish several theoretical results characterizing the problem. These results describe how additional future information affects Bayes classification risk, provide bounds on the classification error of learned Foreclassing models, and demonstrate how future observations used during training can yield finite-sample improvements over conventional classifiers. To provide a benchmark for future research, we propose ForeClassNet, a family of neural network models that jointly perform forecasting and classification while accounting for uncertainty in future outcomes. We investigate a variety of model architectures and evaluate their performance against conventional time series classifiers and forecasting-then-classify approaches. Experiments on datasets spanning weather, energy, and finance demonstrate the potential of Foreclassing to improve classification performance, probabilistic prediction, and downstream decision-making. Together, these theoretical and empirical contributions establish a foundation for future research on Foreclassing.

You can view the paper here.