The Effect of the Prior and the Experimental Design on the Inference of the Precision Matrix in Gaussian Chain Graph Models

TL;DR

This paper explores how experimental design and prior choices influence the accuracy of estimating the precision matrix in Gaussian chain graph models, with implications for biological network inference.

Contribution

It demonstrates that certain priors allow for optimal experimental design, while others do not, providing guidance for designing experiments in biological network analysis.

Findings

Flat and conjugate priors render experimental design ineffective for precision matrix inference.

Normal-MGIG and independent priors enable optimization of experimental design.

Simulation confirms theoretical bounds on information gain from experiments.

Abstract

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.