Thank you. If we specify that KL is continuous at $(S_2, S_1)$ (respectively $(S_3, S_1)$) and that the distributions $S_1$, $S_2$, $S_3$ are strictly positive over all the support elements. Is it possible to characterize $D_{KL}(P_2,P_1)/D_{KL}(P_3,P_1)$ ?

In the case of Bayesian networks: "It has been noted that different Bayesian networks may be equivalent in the sense that they actually represent the same joint probability distribution (and thus conditional independency information as well), even though they have different graphical structures." (cs.uregina.ca/Research/Techreports/2002-02.ps). I am asking the same question for MRFs.

Forgive me. Supposing that we have an infinite network $G_{\infty}$ with vertices $V=[1, \infty]$. Herein, $q_i$ is the value of the node $i \in V$. The $q_i$ are determined by a random walk that starts from a node.