Bayesian framework adapted for neural network learning, Bayesian neural networks, have received widespread attention and successfully applied to various applications. Bayesian inference for neural networks promises improved predictions with reliable uncertainty estimates, robustness, principled model comparison, and decision-making under uncertainty. In this dissertation, we propose novel theoretically consistent Bayesian neural network models and provide their computationally efficient posterior inference algorithms.In Chapter 2, we introduce a Bayesian quantile regression neural network assuming an asymmetric Laplace distribution for the response variable. The normal-exponential mixturere presentation of the asymmetric Laplace density is utilized to derive the Gibbs sampling coupled with Metropolis-Hastings algorithm for the posterior inference. We establish the posterior consistency under a misspecified asymmetric Laplace density model. We illustrate the proposed method with simulation studies and real data examples.Traditional Bayesian learning methods are limited by their scalability to large data and feature spaces due to the expensive inference approaches, however recent developments in variational inference techniques and sparse learning have brought renewed interest to this area. Sparse deep neural networks have proven to be efficient for predictive model building in large-scale studies. Although several works have studied theoretical and numerical properties of sparse neural architectures, they have primarily focused on the edge selection.In Chapter 3, we propose a sparse Bayesian technique using spike-and-slab Gaussian prior to allow for automatic node selection. The spike-and-slab prior alleviates the need of an ad-hoc thresholding rule for pruning. In addition, we adopt a variational Bayes approach to circumvent the computational challenges of traditional Markov chain Monte Carlo implementation. In the context of node selection, we establish the variational posterior consistency together with the layer-wise characterization of prior inclusion probabilities. We empirically demonstrate that our proposed approach outperforms the edge selection method in computational complexity with similar or better predictive performance.The structured sparsity (e.g. node sparsity) in deep neural networks provides low latency inference, higher data throughput, and reduced energy consumption. Alternatively, there is a vast albeit growing literature demonstrating shrinkage efficiency and theoretical optimality in linear models of two sparse parameter estimation techniques: lasso and horseshoe. In Chapter 4, we propose structurally sparse Bayesian neural networks which systematically prune excessive nodes with (i) Spike-and-Slab Group Lasso, and (ii) Spike-and-Slab Group Horseshoe priors, and develop computationally tractable variational inference We demonstrate the competitive performance of our proposed models compared to the Bayesian baseline models in prediction accuracy, model compression, and inference latency.Deep neural network ensembles that appeal to model diversity have been used successfully to improve predictive performance and model robustness in several applications. However, most ensembling techniques require multiple parallel and costly evaluations and have been proposed primarily with deterministic models. In Chapter 5, we propose sequential ensembling of dynamic Bayesian neural subnetworks to generate diverse ensemble in a single forward pass. The ensembling strategy consists of an exploration phase that finds high-performing regions of the parameter space and multiple exploitation phases that effectively exploit the compactness of the sparse model to quickly converge to different minima in the energy landscape corresponding to high-performing subnetworks yielding diverse ensembles. We empirically demonstrate that our proposed approach surpasses the baselines of the dense frequentist and Bayesian ensemble models in prediction accuracy, uncertainty estimation, and out-of-distribution robustness. Furthermore, we found that our approach produced the most diverse ensembles compared to the approaches with a single forward pass and even compared to the approaches with multiple forward passes in some cases.
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