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Register a new userComparing Spike and Slab Priors for Bayesian Variable Selection
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Gertraud Malsiner-Walli
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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An important task in building regression models is to decide which regressors should be included in the final model. In a Bayesian approach, variable selection can be performed using mixture priors with a spike and a slab component for the effects subject to selection. As the spike is concentrated at zero, variable selection is based on the probability of assigning the corresponding regression effect to the slab component. These posterior inclusion probabilities can be determined by MCMC sampling. In this paper we compare the MCMC implementations for several spike and slab priors with regard to posterior inclusion probabilities and their sampling efficiency for simulated data. Further, we investigate posterior inclusion probabilities analytically for different slabs in two simple settings. Application of variable selection with spike and slab priors is illustrated on a data set of psychiatric patients where the goal is to identify covariates affecting metabolism.
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We address the problem of dynamic variable selection in time series regression with unknown residual variances, where the set of active predictors is allowed to evolve over time. To capture time-varying variable selection uncertainty, we introduce new dynamic shrinkage priors for the time series of regression coefficients. These priors are characterized by two main ingredients: smooth parameter evolutions and intermittent zeroes for modeling predictive breaks. More formally, our proposed Dynamic Spike-and-Slab (DSS) priors are constructed as mixtures of two processes: a spike process for the irrelevant coefficients and a slab autoregressive process for the active coefficients. The mixing weights are themselves time-varying and depend on lagged values of the series. Our DSS priors are probabilistically coherent in the sense that their stationary distribution is fully known and characterized by spike-and-slab marginals. For posterior sampling over dynamic regression coefficients, model selection indicators as well as unknown dynamic residual variances, we propose a Dynamic SSVS algorithm based on forward-filtering and backward-sampling. To scale our method to large data sets, we develop a Dynamic EMVS algorithm for MAP smoothing. We demonstrate, through simulation and a topical macroeconomic dataset, that DSS priors are very effective at separating active and noisy coefficients. Our fast implementation significantly extends the reach of spike-and-slab methods to large time series data.
Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation. Specifically, we study the usefulness of continuous bimodal priors to model hypervariance parameters, and the effect scaling has on the posterior mean through its relationship to penalization. Several model selection strategies, some frequentist and some Bayesian in nature, are developed and studied theoretically. We demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification, and show this is achieved using the posterior from a rescaled spike and slab model. We also show how to verify a procedures ability to reduce model uncertainty in finite samples using a specialized forward selection strategy. Using this tool, we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty.
The impracticality of posterior sampling has prevented the widespread adoption of spike-and-slab priors in high-dimensional applications. To alleviate the computational burden, optimization strategies have been proposed that quickly find local posterior modes. Trading off uncertainty quantification for computational speed, these strategies have enabled spike-and-slab deployments at scales that would be previously unfeasible. We build on one recent development in this strand of work: the Spike-and-Slab LASSO procedure of Rov{c}kov{a} and George (2018). Instead of optimization, however, we explore multiple avenues for posterior sampling, some traditional and some new. Intrigued by the speed of Spike-and-Slab LASSO mode detection, we explore the possibility of sampling from an approximate posterior by performing MAP optimization on many independently perturbed datasets. To this end, we explore Bayesian bootstrap ideas and introduce a new class of jittered Spike-and-Slab LASSO priors with random shrinkage targets. These priors are a key constituent of the Bayesian Bootstrap Spike-and-Slab LASSO (BB-SSL) method proposed here. BB-SSL turns fast optimization into approximate posterior sampling. Beyond its scalability, we show that BB-SSL has a strong theoretical support. Indeed, we find that the induced pseudo-posteriors contract around the truth at a near-optimal rate in sparse normal-means and in high-dimensional regression. We compare our algorithm to the traditional Stochastic Search Variable Selection (under Laplace priors) as well as many state-of-the-art methods for shrinkage priors. We show, both in simulations and on real data, that our method fares superbly in these comparisons, often providing substantial computational gains.
We propose a Bayesian procedure for simultaneous variable and covariance selection using continuous spike-and-slab priors in multivariate linear regression models where q possibly correlated responses are regressed onto p predictors. Rather than relying on a stochastic search through the high-dimensional model space, we develop an ECM algorithm similar to the EMVS procedure of Rockova & George (2014) targeting modal estimates of the matrix of regression coefficients and residual precision matrix. Varying the scale of the continuous spike densities facilitates dynamic posterior exploration and allows us to filter out negligible regression coefficients and partial covariances gradually. Our method is seen to substantially outperform regularization competitors on simulated data. We demonstrate our method with a re-examination of data from a recent observational study of the effect of playing high school football on several later-life cognition, psychological, and socio-economic outcomes.
Sparse principal component analysis (PCA) is a popular tool for dimensional reduction of high-dimensional data. Despite its massive popularity, there is still a lack of theoretically justifiable Bayesian sparse PCA that is computationally scalable. A major challenge is choosing a suitable prior for the loadings matrix, as principal components are mutually orthogonal. We propose a spike and slab prior that meets this orthogonality constraint and show that the posterior enjoys both theoretical and computational advantages. Two computational algorithms, the PX-CAVI and the PX-EM algorithms, are developed. Both algorithms use parameter expansion to deal with the orthogonality constraint and to accelerate their convergence speeds. We found that the PX-CAVI algorithm has superior empirical performance than the PX-EM algorithm and two other penalty methods for sparse PCA. The PX-CAVI algorithm is then applied to study a lung cancer gene expression dataset. $mathsf{R}$ package $mathsf{VBsparsePCA}$ with an implementation of the algorithm is available on The Comprehensive R Archive Network.
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