基于SVD和边缘化的快速线性回归
A Fast Linear Regression via SVD and Marginalization
摘要：
我们描述了一个数值方案，用于评估后验矩的贝叶斯线性回归模型与部分池的系数。评价的主要分析工具是从系数空间到预测矩阵的奇异向量空间的基础的变化。在这种变化的基础和分析积分，我们减少了问题的密度超过k + m维的时刻，找到一个m维密度的时刻，其中k是系数的数量和k + m是后验的维度。然后可以使用例如MCMC、梯形规则或自适应高斯求积来计算矩。预测矩阵的SVD的评估是主要的计算成本，并且在预计算阶段执行一次。我们证明了算法的数值结果。本文所描述的方案自然地推广到多水平和多组分层回归模型，其中正态-正态参数出现。
英文摘要：
We describe a numerical scheme for evaluating the posterior moments of Bayesian linear regression models with partial pooling of the coefficients. The principal analytical tool of the evaluation is a change of basis from coefficient space to the space of singular vectors of the matrix of predictors. After this change of basis and an analytical integration, we reduce the problem of finding moments of a density over k + m dimensions, to finding moments of an m-dimensional density, where k is the number of coefficients and k + m is the dimension of the posterior. Moments can then be computed using, for example, MCMC, the trapezoid rule, or adaptive Gaussian quadrature. An evaluation of the SVD of the matrix of predictors is the dominant computational cost and is performed once during the precomputation stage. We demonstrate numerical results of the algorithm. The scheme described in this paper generalizes naturally to multilevel and multi-group hierarchical regression models where normal-normal parameters appear.
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