Bayesian vs. Markov Networks


 Claude Quinn
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1 Bayesian vs. Markov Networks Le Song Machine Learning II: dvanced Topics CSE 8803ML, Spring 2012
2 Conditional Independence ssumptions Local Markov ssumption Global Markov ssumption π ππππππ πππππππ‘π πππ π΄ π΅ πΆ, π πππΊ π΄, π΅; πΆ ππππππ πππππππ‘π πππ π΅π ππ π π» π (π΄ π») π΄ π» π πΉ π΄ π π΄ πΉ (π΄ πΉ π) π π π» π΄ π΅ Derived local and pairwise assumption Dseparation, active trail π΄ πΆ π π» π (π π») π πβππ ππ π‘ ππ΅π π π πβππ ππ π‘ (no X Y) π΄ πΆ π π΅ π· ππ΅π = {π΄π΅πΆπ·} 2
3 Distribution Factorization Bayesian Networks (Directed Graphical Models) πΌ πππ: πΌπ πΊ πΌ π π π(π1,, ππ ) = Conditional Probability Tables (CPTs) π(ππ ππππ ) π=1 Markov Networks (Undirected Graphical Models) π π‘ππππ‘ππ¦ πππ ππ‘ππ£π π, πΌ πππ: πΌ πΊ πΌ π Clique π Potentials 1 π(π1,, ππ ) = Ψπ π·π π Maximal π=1 Normalization (Partition Function) π π = π₯1,π₯2,,π₯π Ψπ π·π Clique π=1 3
4 Representation Power? π ππ π΅π convert? Minimal Imap not unique Do not always have Pmap π1 π1 π3 π2, π4 π2 π4 π1, π3 π4 π2 Minimal Imap unique Do not always have Pmap π΄ πΉ π π΄ πΉ (π΄ πΉ π) π3 4
5 Is there a BN that is a Pmap for a given MN? MN for swing couple of variables does not have a Pmap as BN X 1 X 1 X 1 X 2 X 4 X 2 X 4 X 2 X 4 X 3 X 3 X 3 X 1 X 3 X 2, X 4 X 2 X 4 X 1, X 3 X 1 X 3 X 2, X 4 X 2 X 4 X 1, X 3 X 1 X 3 X 2, X 4 X 2 X 4 X 1, X 3 5
6 Is there an MN that is Pmap for a given BN? BN for Vstructure does not have a Pmap as MN F F F S S S F ( F S) F ( F S) F S F ( F S) 6
7 Conversion using Minimal Imap instead Instead of attempting Pmaps between BNs and MNs, we can try minimal Imaps for conversion Recall: G is a minimal Imap for P if I G I P Removal of a single edge in G render it not an Imap Note: If G is a minimal Imap of P, G need not necessarily satisfy all conditional independence relation in P 7
8 Conversion from BN to MN MN: Markov blanket MB Xi of X i : the set of immediate neighbors of X i in the graph X i V X i MB Xi MB Xi : i Markov blanket for BN? I J B B X E X F C D H C D G MB X =, B, C, D MB X =? (X DF BCD) 8
9 Markov blanket for BN Strategy: go outward from X, try to block all active trails to X 1 2 I B J I B J 3 E X F 4 E X F H C D G H C D G MB X =, B, C, D, E, F X V X MB X MB X 9
10 Markov Blanket for BN MB X in BN is the set of nodes consisting of X s parents, X s children and other parents of X s children Moral graph M(G) of a BN G is an undirected graph that contains an undirected edge between X and Y if There is a directed edge between them in the either direction X and Y are parents of a common children Moral graph insure that MB X in the set of neighbors in undirected graph M(G) B moralize B M G G C D C D 10
11 Minimal Imap from BNs to MNs Moral graph of M G of any BN G is a minimal Imap for G Moralization turns each X, Pa X into a fully connected component CPTs associated with BN can be used as clique potentials E E B moralize B M G G C D C D The moral graph loses some independence relation eg. BN: E, B, E MN: can not read marginal independence 11
12 Perfect Imaps from BNs to MNs If BN G is already moral, then its moral graph M(G) is a perfect Imap of G Proof Sketch: I M G I G The only independence relations that are potentially lost from G to M(G) are those arising from Vstructures Since G has no Vstructures (already moral), no independence are lost in M G 12
13 How about dseparation? Dseparation, active trail S ( H) H S H N S H N H S (N H) S F F ( F S) Judge Dseparation using moral graph Let U = X, Y, Z be three disjoint sets of nodes in a BN G Let G + be the ancestral graph: the induced BN over U ancestors U Then dsep G (X; Y Z) iff sep M(G + )(X; Y Z) G dsep G (; B E) dsep G (; B D, F) C E B F G D B M(G + ) C E sep M(G + )(; B E) B M(G + ) C D F sep M(G + )(; B D, F) 13
14 Why M G + works? Key: Information blocked through common children in G that are not in the conditioning variables, is simulated by ignoring all children X 1 X 1 X 1 X 2 X 4 X 2 X 4 X 2 X 4 X 3 G: X 2 X 4 X 1 X 3 M(G): (X 2 X 4 X 1 ) M(G + ): X 2 X 4 X 1 14
15 Summary: Minimal Imaps from BNs to MNs Moral Graph M(G) is minimal Imap of G If G is already moral, then M(G) is a perfect Imap of G Dsep_G(X; Y Z) sep M(G + ) (X; Y Z) 15
16 Minimal Imaps from MNs to BNs ny BN Imap for an MN must add triangulating edges into the graph Intuition: Vstructures in BN introduce immoralities These immoralities were not present in a Markov networks Triangulation eliminates immoralities X 1 X 1 X 2 X 4 triangulate X 2 X 4 X 3 X 3 16
17 Chordal graphs Let X 1 X 2 X k X 1 be a loop in a graph. chord in a loop is an edge connecting nonconsecutive X i and X j n undirected graph G is chordal if any loop X 1 X 2 X k X 1 for k 4 has a chord B C D E F directed graph G is chordal if its underlying undirected graph is chordal 17
18 Minimal Imaps from MNs to BNs Let H be an MN, and G be any BN minimal Imap for H. Then G can have no immoralities Intuitive reason: immoralities introduce additional independencies that are not in the original MN F Let G any BN minimal Imap for H. Then G is necessarily chordal! Because any nontriangulated loop of length at least 4 in a Bayesian network necessarily contains an immorality S F ( F S) Process of adding edges are called triangulation B D 18 C
19 Minimal Imaps from MNs to BNs Let H be a nonchordal MN. Then there is no BN G that is a perfect Imap for H Proof Sketch Minimal Imap G for H is chordal It must therefore contain additional edges no present in H Each additional edge eliminate some independence assumptions How about perfect Imaps from MNs to BNs? 19
20 Clique trees Notation Let G be a connected undirected graph. Let D 1,, D k be the set of maximal cliques in G Let T be a tree structured graph whose nodes are D 1,, D k Let D i and D j be two cliques in the graph G connected by an edge, Let S ij = D i D j be the separator set between D i and D j Let W ij = D i S ij, the residue set D 1 ={,B,C} D 1 B C S 12 =D 1 D 2 ={B,C} BC D D 2 = {B, C, D} D 2 20
21 Clique trees (cont.) tree T is a clique tree for G if Each node corresponds to a clique in G and each maximal clique in G is a node in T Each separator set S ij separates W ij and W ji Every undirected chordal graph G has a clique tree T Proof by induction (start from an triangle and add nodes) 21
22 Clique tree example BC BC D BC B C D E L BCD CD DCE CE B E CD CEL DE D L CE F DEF C F DE 22
23 Perfect Imaps from MN to BN Let H be a chordal MN. Then there existis a BN G such that I H = I G BC BC 1 D BC Proof Sketch Since H is a chordal MN, is has a clique tree Number the nodes consistent with the clique ordering > 1, B > 2, C> 3 D > 4, E> 5 F > 6, L > 7 BCD CD DCE DE DEF 2 B E CD CE CEL 3 D L CE C F DE
24 Perfect Imaps from MNs to BNs (cont.) Let H be a chordal MN. Then there existis a BN G such that I H = I G Proof sketch (cont.) For each node X i, Let D k be the first clique it occurs in 1 BC BC BCD CD DCE DE DEF CE CEL Define Pa Xi = D k X i X 1,, X i 1 MN H and BN G has the same edges ll parents of each X i are in the same clique They are connected No immoralities in G B2 D4 F6 C3 E5 L7 24
25 Summary: Minimal Imaps from MNs to BNs minimal Imap BN of an MN is chordal Obtained by triangulating the MN If the MN is already chordal, there is a pefect BN Imap for the MN Obtained from the corresponding clique tree 25
26 Partially Directed cyclic Graphs lso called chain graphs (superset of MN and BN) Nodes can be disjointly partition into several chain components n edge within the same chain components must be undirected n edge between two nodes in different chain components be directed B C D E H Chain Components:, B, C, D, E, F, G, H, I F G I 26
27 MN: Gaussian Graphical Models Gaussian distribution can be represented by a fully connected graph with pairwise edge potentials over continuous variable nodes The overall exponential form is: P X 1,, X n = exp ( ij E (X i μ i )Σ 1 ij (X j μ j )) = exp ( X μ Σ 1 X μ ) lso know as Gaussian graphical models (GGM) 27
28 Sparse precision vs. sparse covariance in GGM X 1 X 2 X 3 X 4 X 5 Σ 1 = Σ = Σ 1 15 = 0 X 1 X 5 TheRest X 1 X 5 Σ 15 = 0 28
29 Summary undirected trees undirected chordal graph P BN MN Moralize BN Triangulating MN 29
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