$l_1$-ball Prior: Uncertainty Quantification with Exact Zeros

Lasso and $l_1$-regularization play a dominating role in high dimensional statistics and machine learning. The most attractive property is that it produces a sparse parameter estimate containing exact zeros. For uncertainty quantification, popular Bayesian approaches choose a continuous prior that puts concentrated mass near zero; however, as a limitation, the continuous posterior cannot be exactly sparse. This makes such a prior problematic for advanced models, such as the change-point detection, linear trend filtering and convex clustering, where zeros are crucial for dimension reduction. In this article, we propose a new class of prior, by projecting a continuous distribution onto the $l_1$-ball with radius $r$. This projection creates a positive probability on the lower-dimensional boundary of the ball, where the random variable now contains both continuous elements and exact zeros; meanwhile, assigning prior for radius $r$ gives robustness to large signals. Compared with the spike-and-slab prior, our proposal has substantial flexibility in the prior specification and adaptive shrinkage on small signals; in addition, it enjoys an efficient optimization-based posterior estimation. In asymptotic theory, the prior attains the minimax optimal rate for the posterior concentration around the truth; in practice, it enables a direct application of the rich class of $l_1$-tricks in Bayesian models. We demonstrate its potentials in a data application of analyzing electroencephalogram time series data in human working memory study, using a non-parametric mixture model of linear trend filters.