The tiling by the minimal separators of a junction tree and applications to graphical models Durham, 2008

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1 The tiling by the minimal separators of a junction tree and applications to graphical models Durham, 2008 G. Letac, Université Paul Sabatier, Toulouse. Joint work with H. Massam 1

2 Motivation : the Wishart distributions on decomposable graphs. We denote by S n the space of symmetric real matrices of order n and by P n S n the cone of positive definite matrices. Let G = (V, E) be a decomposable graph with V = {1,..., n}. The subspace ZS G S n is the space of symmetric matrices (s ij ) with zeros prescribed by G, that means s ij = 0 when {i, j} / E. We denote P G = ZS G P n. A space isomorphic to ZS G is the space IS G of symmetric incomplete matrices which are actually real functions on the union of the set V and the set E of edges. 2

3 We denote by π the natural projection of S n on IS G and denote Q G = π(pg 1 ). Three equivalent properties 1. Q G = π(pg 1 ) (definition) 2. Q G is the open convex cone which is the dual of the cone P G. 3. the restriction x C of x Q G to any clique C is positive definite (Hélène s definition). 3

4 Example : If G = the cone P G is the set of positive definite matrices of the form y 1 y 12 0 y 12 y 2 y 23 0 y 32 y 3 The cone Q G is the set of incomplete matrices of the form x 1 x 12 x 12 x 2 x 23 x 32 x 3 such that the two submatrices associated to the two cliques [ ] [ ] x1 x 12 x2 x, 23 x 12 x 2 x 32 x 3 are positive definite. 4

5 The bijection between P G and Q G. Let G decomposable, let C and S be the families of cliques and minimal separators. If x Q G define the Lauritzen function : y = ψ(x) = [x 1 C ] 0 C C S S ν(s)[x 1 S ] 0 where [a] 0 means extension by zeros of a principal submatrix a of S n and where ν(s) is a certain positive integer called multiplicity of S. Theorem 1. The map x y = ψ(x) is a diffeomorphism from Q G onto P G. Its inverse y x from P G onto Q G is x = π(y 1 ). 5

6 Let us fix α : C R and β : S R and let us introduce the function x H(α, β; x) on Q G by H(α, β; x) = C C det(x C ) α(c) S S det(x S ) ν(s)β(s). Define the measure on Q G by µ G (dx) = H( 1 2 ( C +1), 1 2 ( S +1; x)1 Q G (x)dx. 6

7 Perfect orderings of the cliques. Let C be the family of the k cliques of the connected graph (not necessarily decomposable). Consider a bijection P : {1,..., k} C and S P (j) = [P (1) P (2)... P (j 1)] P (j) for j 2. Then the ordering P is said to be perfect if there exists i j < j such that S P (j) P (i j ). This is a deep notion : a connected graph is decomposable if and only if a perfect ordering of the cliques exists. Furthermore if G is decomposable and if P is perfect then S P (j) is a minimal separator. 7

8 Let us fix a perfect ordering P of the set C of the cliques. For a fixed minimal separator S consider the set of cliques J(P, S) = {C C ; j 2 such that P (j) = C et S P (j) = S}. An important result is that if P is a perfect ordering and if for all S S different from S P (2) one has C J(P,S) (α(c) β(s)) = 0 ( we denote by A P this set of (α, β) s) then by a long calculation one sees that there exists a number Γ(α, β) with the following eigenvalue property : for all y P G Q G e tr xy H(α, β; x)µ G (dx) = Γ(α, β)h(α, β; π(y 1 )). (L.-Massam, Ann. Statist. 2007). 8

9 A reformulation is Q G e tr xψ(x 1) H(α, β; x)µ G (dx) = Γ(α, β)h(α, β; x 1 ) namely the functions x H(α, β; x) are eigenfunctions of the operator f K(f) on functions on Q G defined by K(f)(x 1 ) = Q G e tr xψ(x 1) f(x)µ G (dx). 9

10 This leads to the definition of the Wishart distributions on Q G by 1 Γ(α, β)h(α, β; π(y 1 )) e tr (xy) H(α, β; x)µ G (dx) They are therefore indexed by the shape parameters (α, β) and by the scale parameter y P G. There is an other family of similar formulas where the roles of P G and Q G are exchanged that I have no time to describe. 10

11 Homogeneous graphs and the graph A 4. I have to mention that if G is homogeneous, that is if P G is an homogeneous cone (This happens if and only if G does not contains the chain A 4 : as an induced graph), the above formulas hold for a wider range of parameters α and β than the union of A P where P runs all the perfect orderings. Thus the simplest non homogeneous graph is G = A 4 = with cliques and separators C 1 = {1, 2}, C 2 = {2, 3}, C 3 = {3, 4}, S 2 = {2}, S 3 = {3}. An element of Q G has the form x = x 1 x 12 x 21 x 2 x 23 x 32 x 3 x 34 x 43 x 4 for x Q G, with x ij = x ji, 11

12 Let α i = α(c i ), i = 1, 2, 3 β i = β(s i ), i = 2, 3. Define D = {(α, β) α i > 1 2, i = 1, 2, 3, α 1+α 2 > β 2, α 2 +α 3 > β 3 }. Then the following integral (a 7-uple integral!) converges for all σ Q A4 if and only if (α, β) is in D. Under these conditions, it is equal to e x,ψ(σ) H G (α, β; x)µ G (dx) Q G = Γ(α )Γ(α )Γ(α ) Γ(α 2 ) Γ(α 1 + α 2 β 2 )Γ(α 2 + α 3 β 3 ) π 3 2σ α σα 1+α 2 β σ α 2+α 3 β σ α F 1 (α 1 + α 2 β 2, α 2 + α 3 β 3, α 2, σ2 23 σ 2 σ 3 ) where 2 F 1 denotes the hypergeometric function. NB σ i.j means σ i σ ij σj 1 σ ji, thus line 3 is a function of type H(α, β; σ). 12

13 The two lessons of the example A 4 : The integral has the form CH(α, β; σ) if and only if the hypergeometric function degenerates (we mean when c = a or b for 2 F 1 (a, b; c; x). Therefore (α, β) satisfies the eigenvalue property if and only if it is in the union of the A P s for the 4 perfect orderings P of A The 4 perfect orderings are P 1 = C 1 C 2 C 3, P 2 = C 2 C 1 C 3, but P 3 = C 3 C 2 C 1, P 4 = C 2 C 3 C 1 A P1 = A P2 = {α 2 = β 2 } D A P3 = A P4 = {α 3 = β 3 } D Why? As we are going to see, this is because P 1 and P 2 share the same initial minimal separator, as well as P 4 and P 3. 13

14 What one needs to review about decomposable graphs 1. Junction trees. 2. Minimal separators 3. The two definitions of the multiplicity of a minimal separator. 14

15 Junction trees The cliques of a graph are its maximal complete subsets. A junction tree has the set of cliques as set of vertices and is such that if the clique C is on the unique path from C to C then C C C. For instance is a junction tree for the decomposable graph b m v a where the three cliques are 1 = (amu), 2 = (muvc) and 3 = (bmv). A connected graph is decomposable if and only if a junction tree exists (a neat proof of this is given by Blair and Peyton in 1991) u c 15

16 Minimal separators If a and b are not neighbors S V is a separator of a and b if any path from a to b hits S b m v a For instance muvc is a separator of a and b. If nothing can be taken out, S is a minimal separator of a and b. Finally S is minimal separator by itself if there exist non adjacent a and b such that S is a minimal separator of a and b. There are not so many of them, strictly less that the number of cliques anyway. They are mu and mv in the example. A connected graph is decomposable if and only if all the minimal separators are complete (Dirac 1961). u c 16

17 Topological multiplicity of a minimal separator Let S be a minimal separator of a decomposable graph (V, E). Let {V 1,..., V p } be the connected components of V \ S (of course p 2). Let q be the number of j = 1,..., p such that S is NOT a clique of S V j. The number ν(s) = q 1 is called the topological multiplicity of S. Multiplicity of a minimal separator from a perfect ordering If P is a perfect ordering and if S is a minimal separator, denote by ν P (S) the number of j = 2,..., k such that S = S P (j) ; recall S P (j) = [P (1) P (2)... P (j 1)] P (j). (The topological multiplicity is introduced by Lauritzen, Speed and Vivayan in 1984). Question : one observes that in all cases the two definitions of multiplicity coincide. Why? Answer later on. 17

18 Example : If I remove the minimal separator S = 27 to its graph four connected components are obtained : If I add S to each of them, thus for component 1 I obtain the graph for which S = 27 is a clique. This is not the case for the three other connected components 3, 4 et 56. Therefore q = 3, the topological multiplicity of 27 is 2. 18

19 Similarly since the cliques are C 1 = 12, C 2 = 237, C 3 = 247, C 4 = 2567 one can see that P = C 1 C 2 C 3 C 4 is perfect that S P (3) = S P (4) = 27 and that ν P (27) = 2 = ν(p ). Question : Why do we have always ν P (S) = ν(s)? 19

20 Tiling of a junction tree by the minimal separators If (H, E(H)) is a tree (undirected) with vertex set H and edge set E(H) a tiling of H is a family T of subtrees T = {T 1,..., T p } of H such that if E(T i ) is the edge set of T i then {E(T 1 ),..., E(T q )} is a partition of E(H). This implies T 1... T q = H although (T 1,..., T p ) is not a partition of the set H. 20

21 Example g e c h f d b a the tiles of the tiling can be chosen as g h f e f d b c b b a 21

22 Theorem 1. Let G = (V, E) be a decomposable graph and let (C, E(C)) be a junction tree of G. Let S be the family of minimal separators of G. There exists a unique tiling T of the tree (C, E(C)) by subtrees and a bijection S T S from S towards T with the following property : for all S S the edges of T S are the edges {C, C } such that S = C C. Under these circumstances the number of edges of T S is the topological multiplicity of S. Furthermore if C and C are two distinct cliques consider the unique path (C = C 0, C 1,..., C q = C ) from C to C. Let S i S such that {C i 1, C i } is in T Si. Then C C = q i=1 S i. In particular C C = S if C and C are in T S. 22

23 Consider again the example : There are 4 cliques A = {1, 2}, B = {2, 3, 7}, C = {2, 4, 7}, D = {2, 5, 6, 7} and two minimal separators U = {2}, V = {2, 7}. The ordering ABCD of the cliques is perfect with S 2 = U et S 3 = S 4 = V. Thus V has multiplicity 2 and U has multiplicity 1. Consider the junction tree C A B D Then T U = AB et T V = BCD. 23

24 Junction trees and perfect orderings of cliques. Recall that saying that P is a perfect ordering of the set C of the k cliques of a decomposable graph is to say that there exists i j < j such that S P (j) P (i j ). There exist in general several possible i j s. Actually we fix one such i j for each j and we create the graph having C as vertex set with having the k 1 edges {P (i j ), P (j)}. A beautiful result of Beeri, Fagin, Maier and Yannakakis (1983) claims that this graph is a junction tree and conversely that any junction tree can be constructed from a perfect ordering and from a choice of the j i j. Let us say that a junction tree is adapted to the perfect ordering P if there exists a choice j i j giving the tree. 24

25 Tiling by minimal separators and perfect orderings of the cliques. Let P be a perfect ordering of the set C of the k cliques of a decomposable graph Consider now a junction tree adapted to P and let T be the tiling of this tree by the minimal separators. We transform this undirected tree into a rooted tree by taking P (1) as a root. This transforms C into a partially ordered set : C C if the unique path from P (1) to C passes through C. 26

26 Let S be in the set S of the minimal separators. Recall that we have considered before the set of cliques J(P, S) = {C C ; j 2 such that P (j) = C et S P (j) = S}. Just remark that ν P (S) = J(P, S). Now for all S S the subtree T S has a minimal point M(S) for this partial order. Here is now a useful result ruling out the old contest between multiplicities (recall that the number of vertices of a tree is the number of edges plus one ) : 27

27 Theorem 2. J(P, S) = T S \ {M(S)}. In particular ν P (S) is the topological multiplicity T S 1 of S. 28

28 Actually J(P, S) depends on S and on S P (2) only : Theorem 3. Let P and P two perfect orderings such that P (1) P (2) = P (1) P (2), that is to say S P (2) = S P (2) (denoted S 2 ). Then J(P, S) = J(P, S) if S S 2 and J(P, S 2 ) {P (1)} = J(P, S 2 ) {P (1)}. 29

29 Conclusion : Consequences for the Wishart distributions on decomposable graphs. Recall that given a perfect ordering P, the set A P of acceptable shape parameters (α, β) for the Wishart distribution is the set of (α, β) such that for all minimal separators S we have C J(P,S) (α(c) β(s)) = 0. Thus this crucial set A P depends entirely on the family of subsets of cliques F P = {J(P, S); S S}. This tiling process has shown that actually the family F P -and therefore the set A P of shape parameters - depends only on the first minimal separator S P (2) of the perfect ordering P. 30

MATHEMATICAL ENGINEERING TECHNICAL REPORTS. Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph

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