A Noise-Robust Fast Sparse Bayesian Learning Model

This paper utilizes the hierarchical model structure from the Bayesian Lasso in the Sparse Bayesian Learning process to develop a new type of probabilistic supervised learning approach. The hierarchical model structure in this Bayesian framework is designed such that the priors do not only penalize the unnecessary complexity of the model but will also be conditioned on the variance of the random noise in the data. The hyperparameters in the model are estimated by the Fast Marginal Likelihood Maximization algorithm which can achieve sparsity, low computational cost and faster learning process. We compare our methodology with two other popular learning models; the Relevance Vector Machine and the Bayesian Lasso. We test our model on examples involving both simulated and empirical data, and the results show that this approach has several performance advantages, such as being fast, sparse and also robust to the variance in random noise. In addition, our method can give out a more stable estimation of variance of random error, compared with the other methods in the study.Comment: 15 page

Sparse Probit Linear Mixed Model

Linear Mixed Models (LMMs) are important tools in statistical genetics. When used for feature selection, they allow to find a sparse set of genetic traits that best predict a continuous phenotype of interest, while simultaneously correcting for various confounding factors such as age, ethnicity and population structure. Formulated as models for linear regression, LMMs have been restricted to continuous phenotypes. We introduce the Sparse Probit Linear Mixed Model (Probit-LMM), where we generalize the LMM modeling paradigm to binary phenotypes. As a technical challenge, the model no longer possesses a closed-form likelihood function. In this paper, we present a scalable approximate inference algorithm that lets us fit the model to high-dimensional data sets. We show on three real-world examples from different domains that in the setup of binary labels, our algorithm leads to better prediction accuracies and also selects features which show less correlation with the confounding factors.Comment: Published version, 21 pages, 6 figure

The Kernel Interaction Trick: Fast Bayesian Discovery of Pairwise Interactions in High Dimensions

Discovering interaction effects on a response of interest is a fundamental problem faced in biology, medicine, economics, and many other scientific disciplines. In theory, Bayesian methods for discovering pairwise interactions enjoy many benefits such as coherent uncertainty quantification, the ability to incorporate background knowledge, and desirable shrinkage properties. In practice, however, Bayesian methods are often computationally intractable for even moderate-dimensional problems. Our key insight is that many hierarchical models of practical interest admit a particular Gaussian process (GP) representation; the GP allows us to capture the posterior with a vector of O(p) kernel hyper-parameters rather than O(p^2) interactions and main effects. With the implicit representation, we can run Markov chain Monte Carlo (MCMC) over model hyper-parameters in time and memory linear in p per iteration. We focus on sparsity-inducing models and show on datasets with a variety of covariate behaviors that our method: (1) reduces runtime by orders of magnitude over naive applications of MCMC, (2) provides lower Type I and Type II error relative to state-of-the-art LASSO-based approaches, and (3) offers improved computational scaling in high dimensions relative to existing Bayesian and LASSO-based approaches.Comment: Accepted at ICML 2019. 20 pages, 4 figures, 3 table