Clique trees & Belief Propagation. Siamak Ravanbakhsh Winter 2018

Transcription

1 Graphical Models Clique trees & Belief Propagation Siamak Ravanbakhsh Winter 2018

2 Learning objectives message passing on clique trees its relation to variable elimination two different forms of belief propagation

3 Recap: variable elimination (VE) marginalize over a subset - e.g., expensive to calculate (why?) use the factorized form of P P (J) C,D,I,G,S,L,H P (C, D, I, G, S, L, J, H) C,D,I,G,S,L P (D C)P (G D, I)P (S I)P (L G)P (J L, S)P (H G, J)

4 Recap: variable elimination (VE) marginalize over a subset - e.g., P (H, J) expensive to calculate (why?) use the factorized form of P C,D,I,G,S,L P (C, D, I, G, S, L) C,D,I,G,S,L P (D C)P (G D, I)P (S I)P (L G)P (J L, S)P (H G, J) ϕ (H, G, J) 2 note that they do not encode the same CIs think of this as a factor/potential same treatment of Bayes-nets Markov nets for inference

5 Recap: variable elimination (VE) marginalize over a subset - e.g., P (H, J) expensive to calculate (why?) use the factorized form of P C,D,I,G,S,L P (C, D, I, G, S, L) C,D,I,G,S,L ϕ 1(D, C)ϕ 2(G, D, I)ϕ 3(S, I)ϕ 4(L, G)ϕ 5(J, L, S)ϕ 6(H, G, J) =... I ϕ 3(S I) D ϕ 2(G, D, I) C ϕ 1(D, C) repeat this ψ 1 (D) ψ (D, C) 1 =... I ϕ 3(S, I) D ϕ 2(G, D, I)ψ 1 (D) ψ 2 (G, I) ψ 2(G, I, D)

6 Recap: variable elimination (VE) marginalize over a subset - e.g., expensive to calculate (why?) P (J) C,D,I,G,S,L,H P (C, D, I, G, S, L, J, H) eliminate variables in some order C D I

7 Recap: variable elimination (VE) eliminate variables in some order creates a chordal graph maximal cliques are the factors created during VE (ψ ) t chordal graph order: max-cliques C,D,I,H,G,S,L

8 Clique-tree summarize the VE computation using a clique-tree order: C,D,I,H,G,S,L sepset cluster clusters are maximal cliques (factors that are marginalized) C i = Scope[ψ i] P (J) =... I P (S I) D P (G D, I) C P (D C) ψ 1 (D) ψ 1(D, C)

9 Clique-tree summarize the VE computation using a clique-tree order: C,D,I,H,G,S,L sepset cluster clusters are maximal cliques (factors that are marginalized) C i = Scope[ψ i] sepsets are the result of marginalization over cliques S i,j = Scope[ψ i ] S = C C i,j i j

10 Clique-tree: properties a tree T from clusters C i and sepsets S = C C i,j i j family-preserving property: ϕ α(ϕ) = j each factor is associated with a cluster C j s.t. Scope[ϕ] C j

11 Clique-tree: properties a tree T from clusters C i and sepsets S = C C i,j i j family-preserving property: ϕ α(ϕ) = j each factor is associated with a cluster C j s.t. Scope[ϕ] C j running intersection property: if X C, C then for in the path X C k i j C k Ci Cj

12 VE as message passing think of VE as sending messages

13 VE as message passing think of VE as sending messages calculate the product of factors in each clique ψ (C ) i i ϕ:α(ϕ)=i ϕ send messages from the leaves towards a root: δ (S ) = ψ (C ) δ (S ) i j i,j C i S i,j i i k Nb i j k i i,k

14 message passing think of VE as sending messages a different root send messages from the leaves towards a root: δ (S ) = ψ (C ) δ (S ) i j i,j C i S i i i,j k Nb i j k i i,k = V (i j) ϕ F (i j) ϕ the message is the marginal from one side of the tree

15 message passing think of VE as sending messages a different root send messages from the leaves towards a root: δ (S ) ψ (C ) δ (S ) i j i,j C i S i i i,j k Nb i j k i i,k = V (i j) ϕ F (i j) ϕ the belief at the root clique is β r(c r) ψ r(c r) k Nbr δ k r(s r,k) proportional to the marginal β (C ) r r X Ci P (X)

16 message passing: downward pass what if we continue sending messages? (from the root to leaves) root clique i sends a message to clique j when received messages from all the other neighbors k

17 message passing: downward pass what if we continue sending messages? root (from the root to leaves) sum-product belief propagation (BP) δ (S ) = ψ (C ) δ (S ) i j i,j C i S i i i,j k Nb i j k i i,k μ (S ) δ (S )δ (S ) i,j i,j i j i,j j i i,j β (C ) ψ (C ) δ (S ) i i i i k Nbi k i i,k async. message update marginals for any clique (not only root)

18 Clique-tree & queries What type of queries can we answer? marginals over subset of cliques P (A) updating the beliefs after new evidence A C i multiply the (previously calibrated) beliefs propagate to recalibrate (t) β(c )I(E = e ) i (t) P (A E = e ) A C, E C (t) (t) i j

19 Clique-tree & queries What type of queries can we answer? marginals over subset of cliques P (A) updating the beliefs after new evidence multiply the (previously calibrated) beliefs propagate to recalibrate marginals outside cliques: define a super-clique that has both A,B a more efficient alternative? partition function Z P (A, B) A C i (t) P (A E = e ) A C, E C β(c )I(E = e ) i (t) A C i, B Cj (t) (t) i j

20 Chordal graph and clique-tree any chordal graph gives a clique-tree how to get a chordal graph? triangulation use the chordal graph from VE min-neighbor, min-fill... or find the optimal chordal graph smallest tree-width also smallest max-clique NP-hard image: wikipedia

21 Chordal graph and clique-tree Chordal graph = Markov from MRF to Bayes-net: Bayesian networks triangulate build a clique-tree within cliques: fully connected directed edges between cliques: from a root to leaves image: wikipedia

22 Building a clique-tree: example input triangulated clique-tree from: wainwright & jordan

23 clique-tree quiz what clique-tree to use here? what are the sepsets? cost of exact inference?

24 Summary VE as message passing in a clique-tree clique-tree: running intersection & family preserving belief propagation updates: message update belief update types of queries how to build a clique-tree for exact inference

25 bonus slides

26 Clique-tree: calibration represent P using marginals: i i,j E βi μ i,j = i ψi k i δk i = i ψ i = P ~ δ δ i,j E i j j i how about about arbitrary assignments? β i, μi,j i, j E an assignment is calibrated iff BP produces calibrated beliefs can they represent P as above? μ (S ) = β (C ) = β (C ) i,j i,j C S i i,j i i C S j i,j j j for calibrated beliefs these "arbitrary assignments" have to be marginals β (C ) i i,j E i,j i,j P ~ i (X) μ (S ) β (C ) P (C ) i i i

27 BP: an alternative update message update δ (S ) = ψ (C ) δ (S ) i j i,j C i S i i i,j k Nb i j k i i,k calculate the beliefs in the end β (C ) = ψ (C ) δ (S ) i i i i k Nbi k i i,k belief update since δ (S ) = C i S i,j i j i,j δ (S ) j i β (C ) i i,j i we can update the beliefs instead of messages

28 BP: an alternative update belief update initialize β ψ = ϕ, μ 1 i i ϕ:α(ϕ)=i i,j until convergence: pick some (i, j) E μ^i,j C S μ^i,j βj βj μ i,j μ i,j μ^i,j β i i,j i // μ^i,j new i j j i = δ δ μ^i,j δ δ δ δ i j // = old = μ i,j new j i i j j i new δ i j δi j old at convergence, beliefs are calibrated and so they are marginals C i S i,j β (C ) = β (C ) i i C S j i,j j j