  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. Estimation of the precision matrix is a fundamental
task to infer biological graphical structures like microbial networks. We
compare the marginal posterior precision of the precision matrix under four
priors: flat, conjugate Normal-Wishart, Normal-MGIG and a general independent.
Under the flat and conjugate priors, the Laplace-approximated posterior
precision is not a function of the design matrix rendering useless any efforts
to find an optimal experimental design to infer the precision matrix. In
contrast, the Normal-MGIG and general independent priors do allow for the
search of optimal experimental designs, yet there is a sharp upper bound on the
information that can be extracted from a given experiment. We confirm our
theoretical findings via a simulation study comparing i) the KL divergence
between prior and posterior and ii) the Stein's loss difference of MAPs between
random and no experiment. Our findings provide practical advice for domain
scientists conducting experiments to better infer the precision matrix as a
representation of a biological network.
