This work considers variational Bayesian inference as an inexpensive and scalable alternative to a fully Bayesian approach in the context of sparsity-promoting priors. In particular, the priors considered arise from scale mixtures of Normal distributions with a generalized inverse Gaussian mixing distribution. This includes the variational Bayesian LASSO as an inexpensive and scalable alternative to the Bayesian LASSO introduced in [56]. It also includes priors which more strongly promote sparsity. For linear models the method requires only the iterative solution of deterministic least squares problems. Furthermore, for $n\rightarrow \infty$ data points and p unknown covariates the method can be implemented exactly online with a cost of O(p$^3$) in computation and O(p$^2$) in memory. For large p an approximation is able to achieve promising results for a cost of O(p) in both computation and memory. Strategies for hyper-parameter tuning are also considered. The method is implemented for real and simulated data. It is shown that the performance in terms of variable selection and uncertainty quantification of the variational Bayesian LASSO can be comparable to the Bayesian LASSO for problems which are tractable with that method, and for a fraction of the cost. The present method comfortably handles n = p = 131,073 on a laptop in minutes, and n = 10$^5$, p = 10$^6$ overnight.

翻译：这项工作认为,变式贝叶色分解是一种廉价且可缩放的替代方法,可以取代完全巴伊斯办法,在推动迅速的先前期中,特别是,所考虑的前题来自正常分配的比重混合物,一般的高斯混合分布为普遍反向的高斯混合分布,其中包括变式巴伊西亚的LASSO,这是在 [56] 中引入的巴伊西亚的廉价且可缩放的替代巴伊西亚的代谢,还包括更有力地促进恐慌性的前题。对于线性模型来说,该方法仅需要确定性最低方方的迭代解决办法。此外,对于美元(rightrow)\infty$(infty$)的数据点和未知的同位变量,该方法完全可以在线实施,计算成本为O(p_3美元)和O(p_2美元),记忆中的O(p_2美元),这包括:大的近似结果,计算和记忆中的O(p)费用;超度处理器的调整策略也考虑为实际和模拟数据采用的方法。此外,在实际和模拟数据中,在可比较的SAVA值方法中,可变的、可变的、可变的、可变式方法是用于SAIS的、可变式方法。