Independencies in Markov networks
In the previous chapter, we saw how a Bayesian network structure encodes independency conditions in it, and how observing variables affects the flow of influence in the network. Similarly, in the case of Markov networks, the graph structure encodes independency conditions. However, the flow of influence in a Markov network stops as soon as any node is observed in that trail. This is quite different from what we saw in the Bayesian network, where different structures responded differently to the observation of the nodes.
To understand this more formally, let H be a Markov network structure and 
be a set of observed variables. Then, the path 
is active if and only if none of the 
for 
are in Z.
In the case of Bayesian networks, we had the concept of local independencies, where a variable is independent of all its non-descendants, given given its parents. We also had global conditions which were implied by D-Separation. Similarly, in the case of Markov networks...
