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 + 2 dimensions, to finding moments of a 2-dimensional density, where k is the number of coefficients. 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.
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机译：我们描述了一种数值方案，用于评估贝叶斯线性回归模型的后验矩，并对系数进行部分池化。评估的主要分析工具是将基从系数空间更改为预测变量矩阵的奇异向量空间。在这种基数变化和分析积分之后，我们将寻找 k + 2 维密度矩的问题简化为寻找 2 维密度矩的问题，其中 k 是系数数。然后可以使用MCMC、梯形规则或自适应高斯正交等方法计算矩。预测变量矩阵的 SVD 评估是主要的计算成本，在预计算阶段执行一次。我们演示了该算法的数值结果。
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