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Deep neural network (DNN) models have achieved state-of-the-art predictiveaccuracy in a wide range of supervised learning applications. However,accurately quantifying the uncertainty in DNN predictions remains a challengingtask. For continuous outcome variables, an even more difficult problem is toestimate the predictive density function, which not only provides a naturalquantification of the predictive uncertainty, but also fully captures therandom variation in the outcome. In this work, we propose the Bayesian DeepNoise Neural Network (B-DeepNoise), which generalizes standard Bayesian DNNs byextending the random noise variable from the output layer to all hidden layers.The latent random noise equips B-DeepNoise with the flexibility to approximatehighly complex predictive distributions and accurately quantify predictiveuncertainty. For posterior computation, the unique structure of B-DeepNoiseleads to a closed-form Gibbs sampling algorithm that iteratively simulates fromthe posterior full conditional distributions of the model parameters,circumventing computationally intensive Metropolis-Hastings methods. Atheoretical analysis of B-DeepNoise establishes a recursive representation ofthe predictive distribution and decomposes the predictive variance with respectto the latent parameters. We evaluate B-DeepNoise against existing methods onbenchmark regression datasets, demonstrating its superior performance in termsof prediction accuracy, uncertainty quantification accuracy, and uncertaintyquantification efficiency. To illustrate our method's usefulness in scientificstudies, we apply B-DeepNoise to predict general intelligence from neuroimagingfeatures in the Adolescent Brain Cognitive Development (ABCD) project.

Density Regression and Uncertainty Quantification with Bayesian Deep Noise Neural Networks