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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 Steins 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.
The Youden index is a popular summary statistic for receiver operating characteristic curve. It gives the optimal cutoff point of a biomarker to distinguish the diseased and healthy individuals. In this paper, we propose to model the distributions of
In the context of a high-dimensional linear regression model, we propose the use of an empirical correlation-adaptive prior that makes use of information in the observed predictor variable matrix to adaptively address high collinearity, determining i
The notion of exchangeability has been recognized in the causal inference literature in various guises, but only rarely in the original Bayesian meaning as a symmetry property between individual units in statistical inference. Since the latter is a s
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