Bayesian Selection of Graphical Regulatory Models

We de ne a new class of coloured graphical models, called regulatory graphs. These graphs have their own distincitive formal semantics and can directly represent typical qualitative hypotheses about regulatory processes like those described by various biological mechanisms. They admit an embellishment into classes of probabilistic statistical models and so standard Bayesian methods of model selection can be used to choose promising candidate explanations of regulation. 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 through a group structure, it is possible to de ne a conditional conjugate analysis. This means that even when the model space is huge it is nevertheless feasible, using a Bayesian MAP search, to a discover regulatory network with a high Bayes Factor score. We also show that, like the class of Bayesian Networks, regulatory graphs also admit a formal but distinctive causal algebra. The topology of the graph then represents collections of hypotheses about the predicted e ect 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 scienti c interest lies in the circadian regulation of these plants