# Paper No. 12-20

**S Liverani and JQ Smith**

**Bayesian Selection of Graphical Regulatory Models**

**Abstract: **We define a new class of coloured graphical models, called regulatory graphs, which can formally represent typical hypotheses about the dependence structures of regulatory processes like those describing various biological mechanisms. An edge in this graph represents the

hypothesis that the unit representing the donating vertex directly regulates the unit at the receiving vertex. Regulation is modeled by the existence of a deterministic relationship between the longitudinal series of observations labeled by the receiving vertex and the donating one. This class contains longitudinal cluster models as a degenerate graph. Edge colours directly distinguish important features of the mechanism like inhibition and excitation and graphs are often cyclic. With appropriate distributional assumptions, because the regulatory relationships map onto each other by a group structure, it is possible to define a conditional conjugate analysis. This means that even when the model space is huge it is nevertheless feasible, using a Bayesian MAP search, to discover regulatory networks with a high Bayes Factor score. We also show that, like the class of Bayesian Networks, regulatory graphs admit a causal extension. The topology of the graph then represents collections of hypotheses about the predicted effect of controlling the process by tearing out message passers or forcing them to transmit certain signals. We illustrate our methods on a microarray experiment measuring the expression of thousands of genes as a longitudinal series where the scientific interest lies in the circadian regulation of these plants.

**Keywords:** Clustering, gene expression, causal graphs, Bayesian networks, graphical models, regulatory models, MAP model selection