Sum-Product: Message Passing Belief Propagation

Transcription

1 Sum-Product: Message Passing Belief Propagation Probabilistic Graphical Models Sharif University of Technology Spring 2017 Soleymani

2 All single-node marginals If we need the full set of marginals, repeating elimination algorithm for each individual variable is wasteful It does not share intermediate terms Message-passing algorithms on graphs (messages are the shared intermediate terms). sum-product and junction tree upon convergence of the algorithms, we obtain marginal probabilities for all cliques of the original graph. 2

3 Tree Sum-product work only in trees (and we will see it also work on tree-like graphs) Undirected tree A unique path between any pair of nodes Directed tree All nodes have one parent expect to the root 3

4 Parameterization Consider a tree T(V, E) Potential functions: φ x i, φ(x i, x j ) P x = 1 Z i V φ x i i,j E φ x i, x j In directed graphs: P x = P(x r ) P x j x i i,j E φ x r = P(x r ), i r, φ x i = 1 φ x i, x j = P(x j x i ) (x i is the parent of x j ) Z = 1 When we have evidence on variable x i as x i = x i in all factors in which it appears by x i x i we replace 4

5 Sum-product: elimination view Query node r Elimination order: inverse of the topological order Starts from leaves and generates elimination cliques of size at most two Elimination of each node can be considered as messagepassing (or Belief Propagation): Elimination on trees is equivalent to message passing along tree branches Instead of the node elimination, we preserve the node and compute a message from it to its parent This message is equivalent to the factor resulted from the elimination of that node and all of the nodes in its subtree 5

6 Messages root Message that j sends to i 6

7 Messages on a tree Messages can be reused to find probabilities on different query variables. Messages on the tree provide a data structure for caching computations. X 2 We need m 32 (x 2 ) to find both P(X 1 ) and P(X 2 ) X 1 X 3 X 4 X 5 7

8 Messages and marginal distribution Message that X j sends to X i m ji x i = x j φ x j φ x i, x j k N(j)\i m kj (x j ) a function of only x i p x r φ x r k N(r) m kr (x r ) 8

9 Messages and marginal: Example m 12 x 2 = x 1 φ x 1 φ x 1, x 2 p x 2 φ x 2 m 12 (x 2 )m 32 (x 2 )m 42 (x 2 ) 9

10 Computing all node marginals We can compute over all possible elimination order (generating only elimination cliques of size 2) by only computing all possible messages (2 E ) To allow all nodes can be the root, we just need to compute 2 E messages Messages can be reused Instead of running the elimination algorithm N times Dynamic programming approach 2-Pass algorithm that saves and uses messages A pair of messages (one for each direction) have been computed for each edge 10

11 Messages required to compute all node marginals 11

12 A two-pass message-passing schedule Arbitrarily pick a node as the root First pass: starting at the leaves and proceeds inward each node passes a message to its parent. continues until the root has obtained messages from all of its adjoining nodes. Second pass: starting at the root and passing the messages back out messages are passed in the reverse direction. continues until all leaves have received their messages. 12

13 Asynchronous two-pass message-passing 13 First pass: upward Second pass: downward

14 Sum-product algorithm: example m 21 (x 1 ) m 21 (x 1 ) 14

15 Sum-product algorithm: example m 21 (x 1 ) 15

16 Parallel message-passing Message-passing protocol: a node can send a message to a neighboring node when and only when it has received messages from all of its other neighbors Correctness of parallel message-passing on trees The synchronous implementation is non-blocking Theorem: The message-passing guarantees obtaining all marginals in the tree 16

17 Parallel message passing: Example 17

18 Tree-like graphs Sum-product message passing idea can also be extended to work in tree-like graphs (e.g., polytrees) too. Although the undirected marginalized graphs resulted from polytrees are not tree, the corresponding factor graph is a tree 18 Polytree Nodes can have multiple parents Moralized graph Factor graph

19 Recall: Factor graph φ x 1, x 2, x 3 = f a (x 1, x 2 )f b (x 1, x 3 )f c (x 2, x 3 ) φ x 1, x 2, x 3 = f x 1, x 2, x 3 19

20 Sum-product on factor trees Factor tree: a factor graph with no loop Two types of messages: Message that flows from variable node i to factor node s: v is x i = t N i {s} μ ti (x i ) Message that flows from factor node s to variable node i: μ si x i = x N s {i} f s x N(s) j N s {i} v js (x j ) 20

21 Sum-product on factor trees The introduced message-passing schedule for trees can also be used on factor trees When the messages from all the neighbors of a node is received, the marginal probability will be: P x i s N i μ si x i 21 P x i v is (x i )μ si (x i ) s N i s is a factor node that is neighbor of X i

22 The relation between sum-product on factor trees and sum-product on undirected trees Relation of m messages of sum-product algorithm for undirected trees and μ messages of sum-product algorithm for factor trees μ si x i = x N s {i} f s x N(s) j N s {i} v js (x j ) = x j φ(x i, x j )v js (x j ) = φ(x i, x j ) μ tj (x j ) x j t N j {s} = φ(x i )φ(x i, x j ) μ tj (x j ) x j t N j {s} 22 N j = N j {factor corresponding to φ(x j )}

23 Example 23

24 References D. Koller and N. Friedman, Probabilistic Graphical Models: Principles and Techniques, MIT Press, 2009, Chapter 10. M.I. Jordan, An Introduction to Probabilistic Graphical Models, Chapter 4. 24