Bayes Networks Markov Networks Noah Berlow

Bayesian -> Markov (Section 4.5.1) Given B, How can we turn into Markov Network? The general idea: – Convert CPD into factors Given graph G, how can G become H? The general idea: – Convert directed independencies to undirected using Moral Graphs

Background Material: Gibbs Distribution “Borrowed” from Dr. Sridharan’s slides

Distribution Perspective Suppose is a distribution for Bayesian network B with associated graph G The parameterization of B can directly become a Gibbs Distribution. What is the normalization value Z for this factor?

Distribution Example Consider the simple example: B0B1 A0.3.7 A1.1.9 A0A1.4.6 A0,B0A0,B1A1,B0A1,B1.4*.3.4*.7.6*.1.6*.9

Conditioning Perspective Suppose B is conditioned on E = e. Let W = X – {e}. Then is a Gibbs dist. defined by Where each X What is the normalizing factor Z?

Conditioning Example Consider the simple example: B0B1 A0.3.7 A1.1.9 A0A1.4.6 A0,B0A0,B1.4*.3.4*.7

Background Material: Moral Graphs The Moral Graph of G, denoted M[G], is the undirected graph over X that contains undirected edges X – Y if: X -> Y or X <- Y X,Y Pa(Z), Z X BN itself has a Moral graph G if: X,Y Pa(Z) => X->Y or X<-Y

Moral Graph Example DifficultyIntelligence Grade SAT Letter DifficultyIntelligence Grade SAT Letter Job Adapted from Student example in KF book GH

Moral Graphs of G For any distribution s.t. B is a parameterization of G, M[G] is an I-Map for Moreover, M[G] is a minimal I-map for G. (1) This Moral Graph is a Markov network H – Moral Graphs are undirected and encode some of the independencies in the original graph

Sketch of Proof for (1) Markov Blanket for X in G d-separates X from rest of G I.E., No subset of the Markov Blanket has this property. However, information can be lost – V-structures are the biggest culprit – MNs cannot encode V-structure

When are things perfect? If G itself is Moral, then M[G] is a P-map Proof available on pg. 135 However, Moral G is not the norm

Markov -> Bayesian (Section 4.5.2) How can we find a Bayesian Network which is a minimal I-map for a Markov network? “Sadly, it is not quite so easy”

Background Material: Chordal Graphs Chord: In loop a chord is and edge Undirected graph H is chordal if: – has a chord – n > 3

Properties of Markov -> Bayesian If H is a Markov network, the Bayesian network G cannot have immoralities If H is a Markov network, G must be chordal. If H is nonchordal, there is no Bayesian P-map corresponding to H (Theorem 4.11) If H is a chordal Markov network, then there is a Bayesian P-map of H (Theorem 4.13)

Refresher: Active Trails in Bayesian Networks

Markov -> Bayesian Example A CB D F E A CB D F E GH

Resources Koller Friedman Book sections Dr. Sridharan’s class notes