Constructing graphs from distributions

To construct a Markov network from a distribution, the mere concept of I-Maps is not enough. As in the case of Bayesian networks, a fully connected graph has no independence conditions and, hence, it can be an I-Map of any probability distribution. Therefore, we introduce the concept of the minimal I-Map in Markov networks as well. To construct a minimal I-Map, we can use the local independency conditions that we defined in the previous section.

In the first approach, let's consider the case of pairwise independencies. According to pairwise independencies, if there is no edge between {X, Y}, then . Thus, at the very least, to guarantee that H is an I-map, we must add direct edges between all pairs of nodes X and Y, such that they are dependent even on observing all the other variables in the network.

Similarly, we can get more information about the structure by using the local independencies conditions. For each variable X, we can find the minimal set...