On the Equivalence of Directed and Undirected Graphical Models

A directed graphical model can be converted to an undirected graphical model through a process called moralization. Graphically, moralization is accomplished by creating an undirected clique for every node in the directed graph and it's parents.

A directed graph has a conditional probability table (CPT) for every child given its parents. In the undirected graph this CPT becomes the clique potential function, psi. Each node in the undirected graph also has a node potential. If the node had a prior in the directed graph, that becomes the node potential in the undirected graph. Otherwise it is uniformly a constant.

The joint probability function in the directed case goes from:
To the undirected case:

Which translates to: the joint probability of a setting of variables, x, is equal to the product of the clique potential functions evaluated with the relevant elements of x passed as parameters, times the node potential functions of all the individual nodes given their queried setting. This has to be scaled by a normalization factor because the original conditional probability tables, while normalized for a given setting of the parent nodes, are not normalized across the joint distribution of parents and child.

Finally it is worth noting that the moralized graph can represent a larger range of probability distributions than the original directed graph. This might be relevant if you desired to learn the parameters of a directed graph. If you did this in a particular way, namely by moralizing the graph and then learning parameters on that graph, the resulting learned graph might not be representable in the original directed graph.