In the process of solving the actual problem, we often need to convert the directed graph into an undirected graph ), however, the independence between the directed graph with the same structure and the variables that the undirected graph can express is different. How to convert a directed graph into an undirected graph, this undirected graph maximizes the original information, what is the minimum possible loss of the conditional independence contained in the directed graph? Let's start with a straightforward example:
The Joint Distribution of digraph (a) is the product of a series of conditional probability factors: it is very simple to convert to an undirected graph, as shown in (B). Maximal cliques in this undirected graph is a pair of adjacent nodes, therefore, the probability of joint distribution is in the form of absorbing the edge probability of the first node into the first potential function, and so on: Note: Here Partition Function Z = 1.
How can this conversion be extended to other graph structures? In this way, we can convert any digraph-based distribution function into an undirected graph-based factorization.
First, make sure that any variable in the conditional distribution is at least a member of a clique in an undirected graph. For vertices with only one parent node in an undirected graph, you can directly change the directed connection line to the undirected connection line. However, for a node with more than one parent node in a directed graph, you cannot do this again. Is a directed graph with three parent nodes: its joint distribution: The Edge probability is absorbed into the Clique potential, so they belong to the same clique, you need to add a connection line between any two parent nodes. This process of adding a connection line between the parent node is called moralization. The undirected graph is called the moral graph (in this example, the moral graph is fully connected compared with the undirected graph, so it shows that there is no conditional independence. In fact, in the method of converting an undirected graph to a directed graph, the moral graph can maximize the conditional independence of the original directed graph. If we convert a directed graph to an undirected graph, we can simply connect any two nodes to form a fully connected undirected graph. However, we discard all the conditional independence features. However, the moralization process can retain the maximum number of independent features in an undirected graph by adding the least number of connections .)
The undirected graph corresponding to the figure above is as follows:
Summary: The general step of converting a directed graph to an undirected graph is to first add an undirected connection line between the two parent nodes of the directed graph, and then remove the arrow of the original connection line, then we can get the moral graph. Then we need to initialize all clique potential in an undirected graph to 1.
PGM-transition from directed graph to undirected graph (moralization)
