We can represent the joint distribution over a Bayesian network by chain rule, and represent a joint distribution over a Markov network as products of potentials. Then the graph is an I-map of the distribution, and we can use d-separation principles to find conditionally independent variables in the joint distribution.

HMM is a typical directed graph and RBM(Restricted Boltzmann Machine) is a typical undirected graphical model. However, in real applications, sometimes we would make use of hybrid graph models. Taking CRF(Conditional Random Field) as an example. For depth estimation in stereo computer vision, CRF is a competing model, which is a hybrid model, with undirected links among depth values and directed links between depth values and features. So CRF in this case is named as Chain graph. How to represent this hybrid graph using probabilities? Usually, we use products of CRF distributions as its joint probability distribution.

Then how can we determine whether two groups of variable are conditional independent or not? In the chain graph, we will use c-separation principles. By the way, chain graph is a generalization of directed graph and undirected graph. Therefore, probability representation and independence determination in Bayesian Network and Markov Network are specific cases to those in Chain Graph.