Advancing Bayesian deep learning : Sensible priors and accelerated inference

Over the past decade, deep learning has witnessed remarkable success in a wide range of applications, revolutionizing various fields with its unprecedented performance. However, a fundamental limitation of deep learning models lies in their inability to accurately quantify prediction uncertainty, posing challenges for applications that demand robust risk assessment. Fortunately, Bayesian deep learning provides a promising solution by adopting a Bayesian formulation for neural networks. Despite significant progress in recent years, there remain several challenges that hinder the widespread adoption and applicability of Bayesian deep learning. In this thesis, we address some of these challenges by proposing solutions to choose sensible priors and accelerate inference for Bayesian deep learning models. The first contribution of the thesis is a study of the pathologies associated with poor choices of priors for Bayesian neural networks for supervised learning tasks and a proposal to tackle this problem in a practical and effective way. Specifically, our approach involves reasoning in terms of functional priors, which are more easily elicited, and adjusting the priors of neural network parameters to align with these functional priors. The second contribution is a novel framework for conducting model selection for Bayesian autoencoders for unsupervised tasks, such as representation learning and generative modeling. To this end, we reason about the marginal likelihood of these models in terms of functional priors and propose a fully sample-based approach for its optimization. The third contribution is a novel fully Bayesian autoencoder model that treats both local latent variables and the global decoder in a Bayesian fashion. We propose an efficient amortized MCMC scheme for this model and impose sparse Gaussian process priors over the latent space to capture correlations between latent encodings. The last contribution is a simple yet effective approach to improve likelihood-based generative models through data mollification. This accelerates inference for these models by allowing accurate density-esimation in low-density regions while addressing manifold overfitting.