CS281A/Stat241A Lecture 19

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1 CS281A/Stat241A Lecture 19 p. 1/4 CS281A/Stat241A Lecture 19 Junction Tree Algorithm Peter Bartlett

2 CS281A/Stat241A Lecture 19 p. 2/4 Announcements My office hours: Tuesday Nov 3 (today), 1-2pm, in 723 Sutardja Dai Hall. Thursday Nov 5, 1-2pm, in 723 Sutardja Dai Hall. Homework 5 due 5pm Monday, November 16.

3 CS281A/Stat241A Lecture 19 p. 3/4 Key ideas of this lecture Junction Tree Algorithm. (For directed graphical models:) Moralize. Triangulate. Construct a junction tree. Define potentials on maximal cliques. Introduce evidence. Propagate probabilities.

4 CS281A/Stat241A Lecture 19 p. 4/4 Junction Tree Algorithm Inference: Given Graph G = (V, E), Evidence x E, for E V, Set F V, compute p(x F x E ).

5 CS281A/Stat241A Lecture 19 p. 5/4 Junction Tree Algorithm Elimination: Single set F. Any G. Sum-product: All singleton sets F simultaneously. G a tree. Junction tree: All cliques F simultaneously. Any G.

6 CS281A/Stat241A Lecture 19 p. 6/4 Junction Tree Algorithm 1. (For directed graphical models:) Moralize. 2. Triangulate. 3. Construct a junction tree. 4. Define potentials on maximal cliques. 5. Introduce evidence. 6. Propagate probabilities.

7 CS281A/Stat241A Lecture 19 p. 7/4 1. Moralize In a directed graphical model, local conditionals are functions of a variable and its parents: p(x i x π(i) ) = ψ π(i) {i}. To represent this as an undirected model, the set must be a clique. π(i) {i} We consider the moral graph: all parents connected.

8 CS281A/Stat241A Lecture 19 p. 8/4 Junction Tree Algorithm 1. (For directed graphical models:) Moralize. 2. Triangulate. (e.g., run UNDIRECTEDGRAPHELIMINATE) 3. Construct a junction tree. 4. Define potentials on maximal cliques. 5. Introduce evidence. 6. Propagate probabilities.

9 CS281A/Stat241A Lecture 19 p. 9/4 2. Triangulate: Motivation Theorem: A graph G is chordal iff it has a junction tree. Recall that all of the following are equivalent: G is chordal G is decomposable G is recursively simplicial G is the fixed point of UNDIRECTEDGRAPHELIMINATE an oriented version of G has moral graph G. G implies the same cond. indep. as some directed graph.

10 CS281A/Stat241A Lecture 19 p. 10/4 Chordal/Triangulated Definitions: A cycle for a graph G = (V,E) is a vertex sequence v 1,...,v n with v 1 = v n but all other vertices distinct, and {v i,v i+1 } E. A cycle has a chord if it has a pair v i,v j with 1 < i j < n and {i,j} E. A graph is chordal or triangulated if every cycle of length at least four has a chord.

11 CS281A/Stat241A Lecture 19 p. 11/4 Junction Tree: Definition A clique tree for a graph G = (V,E) is a tree T = (V T,E T ) where V T is a set of cliques of G, V T contains all maximal cliques of G. A junction tree for a graph G is a clique tree T = (V T,E T ) for G that has the running intersection property: for any cliques C 1 and C 2 in V T, every clique on the path connecting C 1 and C 2 contains C 1 C 2.

12 CS281A/Stat241A Lecture 19 p. 12/4 Junction Tree: Example C 2 C 4 3 C 3 3 C 1 C 5 Clique Tree: C 1 C 2 C 3 C 4 C 5 C 6 C 6 3

13 CS281A/Stat241A Lecture 19 p. 13/4 2. Triangulate Theorem: A graph G is chordal iff it has a junction tree. Chordal Recursively Simplicial, and we ll show: Recursively Simplicial Junction Tree Chordal

14 CS281A/Stat241A Lecture 19 p. 14/4 Recursively Simplicial Junction Tree Recall: A graph G is recursively simplicial if it contains a simplicial vertex v (neighbors form a clique), and when v is removed, the remaining graph is recursively simplicial. Proof idea induction step: Consider a recursively simplicial graph G of size n + 1. When we remove a simplicial vertex v, it leaves a subgraph G with a junction tree T. Let N be the clique in T containing v s neighbors. Let C be a new clique of v and its neighbors. To obtain a junction tree for G: If N contains only v s neighbors, replace it with C. Otherwise, add C with an edge to N.

15 CS281A/Stat241A Lecture 19 p. 15/4 Junction Tree Chordal Proof idea induction step: Consider a junction tree T of size n + 1. Consider a leaf C of T and its neighbor N in T. Remove C from T and remove C \ N from V. The remaining tree is a junction tree for the remaining (chordal) graph. All cycles have a chord, since either 1. Cycle is completely in remaining graph, 2. Cycle is completely in C, or 3. Cycle is in C \ N, C N, and V \ C (and C N is complete, so contains a chord).

16 CS281A/Stat241A Lecture 19 p. 16/4 Junction Tree Algorithm 1. (For directed graphical models:) Moralize. 2. Triangulate. 3. Construct a junction tree: Find a maximal spanning tree. 4. Define potentials on maximal cliques. 5. Introduce evidence. 6. Propagate probabilities.

17 CS281A/Stat241A Lecture 19 p. 17/4 Junction Tree is Maximal Spanning Tree Define: The weight of a clique tree T = (C,E) is w(t) = C C. (C,C ) E A maximal spanning tree for a clique set C of a graph is a clique tree T = (C,E ) with maximal weight over all clique trees of the form (C,E).

18 CS281A/Stat241A Lecture 19 p. 18/4 Junction Tree is Maximal Spanning Tree Theorem: Suppose that a graph G with clique set C has a junction tree. Then a clique tree (C,E) is a junction tree iff it is a maximal spanning tree for C. And there are efficient greedy algorithms for finding the maximal spanning tree: While graph is not connected: Add an edge for the biggest weight separating set that does not lead to a cycle.

19 CS281A/Stat241A Lecture 19 p. 19/4 Maximal Spanning Tree: Proof Consider a graph G with vertices V. For a clique tree (C,E), define the separators S = { C C : (C,C ) E }. Since T is a tree, for any k V, S S 1[k S] C C 1[k C] 1, with equality iff the subgraph of T of nodes containing k is a tree.

20 CS281A/Stat241A Lecture 19 p. 20/4 Maximal Spanning Tree: Proof Thus, w(t) = S S S = 1[k S] S S k ( ) 1[k C] 1 k C C = C n, C C with equality iff T is a junction tree.

21 CS281A/Stat241A Lecture 19 p. 21/4 Junction Tree Algorithm 1. (For directed graphical models:) Moralize. 2. Triangulate. 3. Construct a junction tree 4. Define potentials on maximal cliques. 5. Introduce evidence. 6. Propagate probabilities.

22 CS281A/Stat241A Lecture 19 p. 22/4 Define potentials on maximal cliques To express a product of clique potentials as a product of maximal clique potentials: ψ S1 (x S1 ) ψ SN (x SN ) = C C ψ C (x C ), 1. Set all clique potentials to For each potential, incorporate (multiply) it into a potential containing its variables.

23 CS281A/Stat241A Lecture 19 p. 23/4 Junction Tree Algorithm 1. (For directed graphical models:) Moralize. 2. Triangulate. 3. Construct a junction tree 4. Define potentials on maximal cliques. 5. Introduce evidence. 6. Propagate probabilities.

24 CS281A/Stat241A Lecture 19 p. 24/4 Introduce Evidence Two equivalent approaches: 1. Introduce evidence potentials δ(x i, x i ) for i in E, so that marginalizing fixes x E = x E. 2. Take slice of each clique potential: ( ) ψ C (x C ) := ψ C xc (V \E), x C E.

25 CS281A/Stat241A Lecture 19 p. 25/4 Junction Tree Algorithm 1. (For directed graphical models:) Moralize. 2. Triangulate. 3. Construct a junction tree 4. Define potentials on maximal cliques. 5. Introduce evidence. 6. Propagate probabilities.

26 CS281A/Stat241A Lecture 19 p. 26/4 Hugin Algorithm Add potentials for separator sets. Recall: If G has a junction tree T = (C, S), then any probability distribution that satisfies the conditional independences implied by G can be factorized as p(x) = C C p(x C) S S p(x S), where, if S = {C 1,C 2 } then x S denotes x C1 C 2.

27 CS281A/Stat241A Lecture 19 p. 27/4 Hugin Algorithm Represent p with potentials on separators: p(x) = C C ψ C(x C ) S S φ S(x S ) This can represent any distribution that an undirected graphical model can (since we can set φ S 1). But nothing more (since we can incorporate each φ S into one of the ψ C s that have S C).

28 CS281A/Stat241A Lecture 19 p. 28/4 Hugin Algorithm 1. Initialize: ψ C (x C ) = appropriate clique potential φ S (x S ) = Update potentials so that p(x) is invariant ψ C,φ S become the marginal distributions.

29 CS281A/Stat241A Lecture 19 p. 29/4 Algorithm: Messages are passed between cliques in the junction tree. A message is passed from V to adjacent vertex W once V has received messages from all its other neighbors. The message corresponds to the updates: φ (1) S (x S) = x V S ψ V (x V ), W (x W) = ψ W (x W ) φ(1) S (x S) φ S (x S ), ψ (1) ψ (1) V (x V ) = ψ V (x V ), where S = V W is the separator.

30 CS281A/Stat241A Lecture 19 p. 30/4 Analysis: 1. p(x) is invariant under these updates. 2. In a tree, messages can path in both directions over every edge. 3. The potentials are locally consistent after messages have passed in both directions. 4. In a graph with a junction tree, local consistency implies global consistency.

31 CS281A/Stat241A Lecture 19 p. 31/4 Analysis: p(x) = C C ψ C(x C ) S S φ S(x S ). In this ratio, only ψ V ψ W φ S changes, to ψ (1) V ψ(1) W φ (1) S = ψ V ψ W φ (1) S φ (1) S φ S = ψ V ψ W φ S.

32 CS281A/Stat241A Lecture 19 p. 32/4 Analysis: 1. p(x) is invariant under these updates. 2. In a tree, messages can path in both directions over every edge. 3. The potentials are locally consistent after messages have passed in both directions. 4. In a graph with a junction tree, local consistency implies global consistency.

33 CS281A/Stat241A Lecture 19 p. 33/4 Analysis: Local Consistency Suppose that messages pass both ways, from V to W and back: 1. a message passes from V to W : φ (1) S (x S) = x V S ψ V (x V ), W (x W) = ψ W (x W ) φ(1) S (x S) φ S (x S ), ψ (1) ψ (1) V (x V ) = ψ V (x V ),

34 CS281A/Stat241A Lecture 19 p. 34/4 Analysis: Local Consistency 2. other messages pass to W : φ (2) S ψ (2) V = φ(1) S = ψ(1) V ψ (2) W =

35 CS281A/Stat241A Lecture 19 p. 35/4 Analysis: Local Consistency 3. a message passes from W to V : φ (3) S (x S) = x W S ψ (2) W (x W), ψ (3) V (x V ) = ψ (2) V (x V ) φ(3) S (x S) φ (2) S (x S), ψ (3) W (x V ) = ψ W (x W ). Subsequently, φ S,ψ V,ψ W remain unchanged.

36 CS281A/Stat241A Lecture 19 p. 36/4 Analysis: Local Consistency After messages have passed in both directions, these potentials are locally consistent: x V \S ψ (3) V (x V ) = x W \S ψ (3) W (x W) = φ (3) S (x S). c.f.: x V \S p(x V ) = x W \S p(x W ) = p(x S ).

37 CS281A/Stat241A Lecture 19 p. 37/4 Analysis: Local Consistency Proof ψ (3) V (x V ) = ψ (2) V (xv ) φ(3) S (x S) x V \S x V \S φ (2) S (x S) = φ(3) S (x S) φ (1) S (x S) (W V ) x V \S ψ V (x V ) (to W ) = φ (3) S (x S) (V W ) = x W \S ψ (2) W (x W) = x W \S ψ (3) W (x W).

38 CS281A/Stat241A Lecture 19 p. 38/4 Analysis: 1. p(x) is invariant under these updates. 2. In a tree, messages can path in both directions over every edge. 3. The potentials are locally consistent after messages have passed in both directions. 4. In a graph with a junction tree, local consistency implies global consistency.

39 CS281A/Stat241A Lecture 19 p. 39/4 Analysis: Local implies Global Local consistency: For all adjacent cliques V,W with separator S, x V \S ψ (3) V (x V ) = x W \S ψ (3) W (x W) = φ (3) S (x S). Global consistency: For all cliques C, ψ C (x C ) = x Cc C ψ C(x C ) S φ S(x S ) = p(x C)

40 CS281A/Stat241A Lecture 19 p. 40/4 Local implies Global: Proof Induction on number of cliques. Trivial for one clique. Assume true for all junction trees with n cliques. Consider a tree T W of size n + 1. Fix a leaf B, attached by separator R. Define W = all variables N = B \ R V = W \ N T V = junction tree without B.

41 CS281A/Stat241A Lecture 19 p. 41/4 Local implies Global: Proof The junction tree T V : Has only variables V (from the junction tree property). Satisfies local consistency. Hence satisfies global consistency for V : for all A T V, ψ A (x A ) = C C V ψ C (x C ) x V S S \A V φ S (x S ).

42 CS281A/Stat241A Lecture 19 p. 42/4 Local implies Global: Proof But viewing this A in the larger T W, we have C C W ψ C (x C ) S S W φ S (x S ) = ψ B (x B ) φ x V \A x R (x R ) N x W \A = x V \A = ψ A (x A ). C C V ψ C (x C ) S S V φ S (x S ) C C V ψ C (x C ) S S V φ S (x S )

43 CS281A/Stat241A Lecture 19 p. 43/4 Local implies Global: Proof And for the new clique B: Suppose A is the neighbor of B in T W. C C W ψ C (x C ) x W \B S S W φ S (x S ) = ψ B (x B ) φ x A\R x R (x R ) V \A = ψ B(x B ) φ R (x R ) x V \A = ψ B (x B ) = ψ B (x B ). x A\R x A\R ψ A (x A ) φ R (x R ) C C V ψ C (x C ) S S V φ S (x S ) C C V ψ C (x C ) S S V φ S (x S )

44 CS281A/Stat241A Lecture 19 p. 44/4 Junction Tree Algorithm 1. (For directed graphical models:) Moralize. 2. Triangulate. 3. Construct a junction tree 4. Define potentials on maximal cliques. 5. Introduce evidence. 6. Propagate probabilities.

45 CS281A/Stat241A Lecture 19 p. 45/4 Junction Tree Algorithm: Computation Size of largest maximal clique determines run-time. If variables are categorical and potentials are represented as tables, marginalizing takes time exponential in clique size. Finding a triangulation to minimize the size of the largest clique is NP-hard.

46 CS281A/Stat241A Lecture 19 p. 46/4 Key ideas of this lecture Junction Tree Algorithm. (For directed graphical models:) Moralize. Triangulate. Construct a junction tree. Define potentials on maximal cliques. Introduce evidence. Propagate probabilities.

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