Variational logistic regression is a popular method for approximate Bayesian inference seeing wide-spread use in many areas of machine learning including: Bayesian optimization, reinforcement learning and multi-instance learning to name a few. However, due to the intractability of the Evidence Lower Bound, authors have turned to the use of Monte Carlo, quadrature or bounds to perform inference, methods which are costly or give poor approximations to the true posterior. In this paper we introduce a new bound for the expectation of softplus function and subsequently show how this can be applied to variational logistic regression and Gaussian process classification. Unlike other bounds, our proposal does not rely on extending the variational family, or introducing additional parameters to ensure the bound is tight. In fact, we show that this bound is tighter than the state-of-the-art, and that the resulting variational posterior achieves state-of-the-art performance, whilst being significantly faster to compute than Monte-Carlo methods.

MOTIVATION: Few Bayesian methods for analyzing high-dimensional sparse survival data provide scalable variable selection, effect estimation and uncertainty quantification. Such methods often either sacrifice uncertainty quantification by computing maximum a posteriori estimates, or quantify the uncertainty at high (unscalable) computational expense. RESULTS: We bridge this gap and develop an interpretable and scalable Bayesian proportional hazards model for prediction and variable selection, referred to as sparse variational Bayes. Our method, based on a mean-field variational approximation, overcomes the high computational cost of Markov chain Monte Carlo, whilst retaining useful features, providing a posterior distribution for the parameters and offering a natural mechanism for variable selection via posterior inclusion probabilities. The performance of our proposed method is assessed via extensive simulations and compared against other state-of-the-art Bayesian variable selection methods, demonstrating comparable or better performance. Finally, we demonstrate how the proposed method can be used for variable selection on two transcriptomic datasets with censored survival outcomes, and how the uncertainty quantification offered by our method can be used to provide an interpretable assessment of patient risk. AVAILABILITY AND IMPLEMENTATION: our method has been implemented as a freely available R package survival.svb (https://github.com/mkomod/survival.svb). SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.