Figure 2

Bayesian mixture of factor analyzers (MFA) model applied to transcriptional regulation. The figure shows a probabilistic independence graph of the Bayesian mixture of factor analyzers (MFA) model proposed in Section 3. Variables are represented by circles, and hyperparameters are shown as square boxes in the graph. components (factor analyzers), each with their own parameters and , are used to model the expression profiles and TF binding profiles of genes. The factor loadings have a zero-mean Gaussian prior distribution, whose precision hyperparameters are given a gamma distribution determined by and . The analyzer displacements and have Gaussian priors determined by the hyperparameters and , respectively. The indicator variables select one out of factor analyzers, and the associated latent variables or factors have normal prior distributions. The indicator variables are given a multinomial distribution, whose parameter vector , the so-called mixture proportions, have a conjugate Dirichlet prior with hyperparameters . and are the diagonal covariance matrices of the Gaussian noise in the expression and binding profiles, respectively. A dashed rectangle denotes a plate, that is an iid repetition over the genes or the mixture components , respectively. The biological interpretation of the model is as follows. represents the composition of the th transcriptional module, that is, it indicates which TFs bind cooperatively to the promoters of the regulated genes. allows for perturbations that result, for example, from the temporary inaccessibility of certain binding sites or a variability of the binding affinities caused by external influences. is the background gene expression profile. represents the activity profile of the th transcriptional module, which modulates the expression levels of the regulated genes. describes the gene-specific susceptibility to transcriptional regulation, that is, to what extent the expression of the th gene is influenced by the binding of a transcriptional module to its promoter. A complete description of the model can be found in Section 3.