Bayesian High-dimensional Linear Regression with Sparse Projection-posterior

TL;DR

This work develops a novel Bayesian framework for high-dimensional linear regression under sparsity by transforming a dense Gaussian posterior into a sparse projection-posterior via an immersion map. The authors establish that the induced sparse posterior contracts at the optimal sparse rate, achieves sign-consistent model selection, and yields credible regions with asymptotically correct frequentist coverage, including a post-selection credible ellipsoid. A key feature is the ability to distribute computation across multiple machines, enabling scalable analysis of very large $p$ while maintaining statistical guarantees. The approach is validated through extensive simulations and real-data analyses (including ADNI), and is implemented in the R package sparseProj to facilitate practical use.

Abstract

We consider a novel Bayesian approach to estimation, uncertainty quantification, and variable selection for a high-dimensional linear regression model under sparsity. The number of predictors can be nearly exponentially large relative to the sample size. We put a conjugate normal prior initially disregarding sparsity, but for making an inference, instead of the original multivariate normal posterior, we use the posterior distribution induced by a map transforming the vector of regression coefficients to a sparse vector obtained by minimizing the sum of squares of deviations plus a suitably scaled $\ell_1$-penalty on the vector. We show that the resulting sparse projection-posterior distribution contracts around the true value of the parameter at the optimal rate adapted to the sparsity of the vector. We show that the true sparsity structure gets a large sparse projection-posterior probability. We further show that an appropriately recentred credible ball has the correct asymptotic frequentist coverage. Finally, we describe how the computational burden can be distributed to many machines, each dealing with only a small fraction of the whole dataset. We conduct a comprehensive simulation study under a variety of settings and found that the proposed method performs well for finite sample sizes. We also apply the method to several real datasets, including the ADNI data, and compare its performance with the state-of-the-art methods. We implemented the method in the \texttt{R} package called \texttt{sparseProj}, and all computations have been carried out using this package.

Figures (9)

• Figure 1: Boxplots of the MSEs for different methods averaged over $M = 100$ replications corresponding to the independent design case.
• Figure 2: Boxplots of the MSEs for different methods averaged over $M = 100$ replications corresponding to the correlated design case.
• Figure 3: Boxplots of the MSEs for different methods averaged over $M = 100$ replications corresponding to the highly correlated design case.
• Figure 4: The average coverage probabilities for the 10 signals are plotted against the average lengths of the confidence/credible intervals for the competing methods.
• Figure 5: Plots of the randomly selected signal and noise coefficients for different methods corresponding to the independent design ($p > n$: lower left panel) and the correlated design ($p > n$: lower right panel) and similarly, the independent design ($p < n$: upper right panel) and the correlated design ($p < n$: upper left panel).

Theorems & Definitions (23)

• Definition 3.1
• Theorem 3.1: Estimation and prediction rate
• Theorem 3.2
• Theorem 3.3: Bernstein-von Mises Theorem
• Corollary 3.3.1
• Theorem 3.4
• Lemma 10.1
• proof
• Lemma 10.2
• proof