Here, we investigate whether (and how) experimental design could aid in the estimation of the precision matrix in a Gaussian chain graph model, especially the interplay between the design, the effect of the experiment and prior knowledge about the effect. We approximate the marginal posterior precision of the precision matrix via Laplace approximation under different priors: a flat prior, the conjugate prior Normal-Wishart, the unconfounded prior Normal-Matrix Generalized Inverse Gaussian (MGIG) and a general independent prior. We show that the approximated posterior precision is not a function of the design matrix for the cases of the Normal-Wishart and flat prior, but it is for the cases of the Normal-MGIG and the general independent prior. However, for the Normal-MGIG and the general independent prior, we find a sharp upper bound on the approximated posterior precision that does not involve the design matrix which translates into a bound on the information that could be extracted from a given experiment. We confirm the theoretical findings via a simulation study comparing the Stein's loss difference between random versus no experiment (design matrix equal to zero). Our findings provide practical advice for domain scientists conducting experiments to decode the relationships between a multidimensional response and a set of predictors.

翻译：在此,我们调查实验设计是否(以及如何)有助于估计高斯链图模型中的精确矩阵,特别是设计、实验效果和以前对影响的认识之间的相互作用。我们根据不同的前科,通过Lappl近似不同前科,将精确矩阵的边缘后部精确度大致地放在不同的前科之下:平坦的前科,前常态-Wishart的共和前科,前常态-马特克一般反面格(MGIG)和一般独立的前科。我们表明,近似后部精确度不是正常-Wishart和平面前情况的设计矩阵的函数,而是普通-MGIG和一般以前独立的情况。然而,对于普通-MGIG和一般前科,我们发现,近似的后部精确度与设计矩阵的紧紧紧相连,而设计矩阵并不包含从特定实验中提取的信息的界限。我们通过模拟研究确认理论结论,将斯坦因随机与无实验而无源的亏损差(设计矩阵等于一个实际的矩阵与一个实际的轨道之间的预测关系)。我们为进行实际的实验提供建议。