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We study the Bayesian multi-task variable selection problem, where the goal is to select activated variables for multiple related data sets simultaneously. We propose a new variational Bayes algorithm which generalizes and improves the recently developed “sum of single effects” model of Wang et al. (2020a). Motivated by differential gene network analysis in biology, we further extend our method to joint structure learning of multiple directed acyclic graphical models, a problem known to be computationally highly challenging. We propose a novel order MCMC sampler where our multi-task variable selection algorithm is used to quickly evaluate the posterior probability of each ordering. Both simulation studies and real gene expression data analysis are conducted to show the efficiency of our method. Finally, we also prove a posterior consistency result for multi-task variable selection, which provides a theoretical guarantee for the proposed algorithms. Supplementary materials for this article are available online. 

We propose an empirical Bayes formulation of the structure learning problem, where the prior specification assumes that all node variables have the same error variance, an assumption known to ensure the identifiability of the underlying causal directed acyclic graph. To facilitate efficient posterior computation, we approximate the posterior probability of each ordering by that of a best directed acyclic graph model, which naturally leads to an order-based Markov chain Monte Carlo algorithm. Strong selection consistency for our model in high-dimensional settings is proved under a condition that allows heterogeneous error variances, and the mixing behaviour of our sampler is theoretically investigated. Furthermore, we propose a new iterative top-down algorithm, which quickly yields an approximate solution to the structure learning problem and can be used to initialize the Markov chain Monte Carlo sampler. We demonstrate that our method outperforms other state-of-the-art algorithms under various simulation settings, and conclude the paper with a single-cell real-data study illustrating practical advantages of the proposed method. 

Geoid is a key observable for understanding the dynamics of the deep Earth, but has been considered largely transparent to long‐wavelength shallow density structures, especially those of the cratonic lithosphere. Here, we demonstrate that the observed flat craton‐ocean geoid pattern, traditionally interpreted as reflecting neutrally buoyant cratonic keels, provides critical constraints on both the net buoyancy and the depth‐dependent density distribution of cratonic mantle lithosphere. Using both simple theoretical calculations and quantitative numerical models, we show that the recent seismic data on lithospheric structure require the existence of a dense cratonic mantle lithosphere to explain the observed topography and geoid. In practice, topography reveals the net buoyancy of the cratonic lithosphere, while geoid further delineates the depth‐dependence of excess density. We find that the mantle lithosphere below large cratons bears net negative buoyancy close to that of a pure‐thermal lithosphere, with most of the excess density distributed within the lower half of the lithosphere. Density profiles of small cratons, due to strong edge effects from surrounding orogenic belts, are harder to constrain, except that their mantle lithosphere is also negatively buoyant.