Conversion Between Markov and Bayesian Networks

In the last few articles, we have looked at various subjects regarding both Bayesian and Markov Networks. We have seen how they can each represent independence constraints that the other cannot. In this article, however, we want to check whether and how it is possible to go from one to another.

From a Bayesian Network to a Markov Network

We can start by considering how we can go from a Bayesian Network to a Markov Network. We can start by remembering that a Bayesian Network is a directed graphical model, which encodes various independence assumptions. Now, there are two possible manners to view this problem:

• Method 1: Given a Bayesian network, B, we can ask how to represent the underlying distribution, P_B, as a parameterized Markov network. In other words, we want to find a minimal I-map for a distribution P_B.
• Method 2: Given a graph, G, we can ask how to represent the independencies in G using an undirected graph, H. In other words, we want to find a minimal I-map for the independencies I(G)

Let us start by considering method 1, i.e., we begin by considering a distribution, P_B, where B is a parameterized Bayesian Network over a graph, G. The interesting about this is that we can see this parameterization of B as a parameterization of a Gibbs distribution, which we remember is how we parameterized a Markov Network. Now, how is this possible? We can simply see the conditional probability…