Belief Updating in Bayes Networks (1)

Transcription

1 Belief Updating in Bayes Networks (1) Mark Fishel, Mihkel Pajusalu, Roland Pihlakas Bayes Networks p. 1

2 Some Concepts Belief: probability Belief updating: probability calculating based on a BN (model, parameters and/or evidences) Belief propagation: algorithm for computing marginals of functions on a graphical model Bayes Networks p. 2

3 Summary Main task: having a bayes network, calculate P(A) (distribution) for each of its variables having some evidence Will introduce a useful representation for the task (domain graphs) Will build theory around the task (triangulated graphs, join trees) Will introduce an uglyness (junction trees) that enables solving the task rather efficiently Bayes Networks p. 3

4 Potentials Conditional probability P(A B,C) : {A,B,C} [0, 1] In the following we only need the domains (the variables), regardless of the order, so P(A B,C) = ϕ(a,b,c) ( potential ) ϕ(a) =< ϕ(a 1 ),ϕ(a 2 ),...,ϕ(a n ) > 1 =< 1, 1,...,1 > (unit potential) (ϕ ψ)(a,b,c,d) = ϕ(a,b,c) ψ(b,d) Potential of n variables n-dimensioned matrix Potential multiplication element-wise Bayes Networks p. 4

5 Example For the BN on the blackboard: ϕ 1 = P(A 1 ), ϕ 2 = P(A 2 A 1 ), ϕ 3 = P(A 3 A 1 ), ϕ 4 = P(A 4 A 2 ), P(A 4 ) P(A 4 ) with e ϕ 5 = P(A 5 A 2,A 3 ) ϕ 6 = P(A 6 A 3 ) main idea: eliminating variables How to compute P(X) efficiently (in which order to eliminate variables)? How to compute marginals for all variables efficiently? Bayes Networks p. 5

6 Domain graph = moral graph of the BN = losing directions + interconnecting all vars of each domain (making cliques) eliminating variables fill-ins complete neighbour set variables (no fill-ins) ( clique outside part, sisepunkt) Bayes Networks p. 6

7 Eliminating in domain graphs perfect elimination sequence: no fill-ins domain set of an elimination sequence is same for all perfect elimination sequences is the set of domain graph cliques Bayes Networks p. 7

8 Triangulated graph = a graph that has a perfect elimination sequence a node with a complete neighbour set simplical node sisepunkt path to any other node goes through its clique Example of a nontriangulated graph Bayes Networks p. 8

9 Eliminating if X 1,...,X k is a perfect elimination sequence and X j is simplical then X j,x 1,...,X j 1,X j+1,...,x k is perfect too NB1: eliminating a simplical node from a triangulated graph results in a triangulated graph NB2: a triangulated graph has at least two simplical nodes NB3: an undirected graph is triangulated all nodes can be eliminated by consequently eliminating simplical nodes Bayes Networks p. 9

10 Cliques Find set of cliques in a triangulated graph: eliminate simplical X; push f a(x) to clique candidate set repeat until f a(x) includes all graph nodes prune the candidate set by excluding subsets of other candidates Bayes Networks p. 10

11 Join trees = tree of cliques where for each pair of nodes V,W all the nodes on the path between them contain V W NB1: cliques of a graph organizable into a join tree graph is triangulated Bayes Networks p. 11

12 Building join trees find simplical X, remove all sisepunktid from same clique (removed i nodes) set of remaining nodes from fa(x) separator V i := fa(x) separator S i iterate until empty graph connect each S i to V j where j > i and S i V j Bayes Networks p. 12

13 Junction trees = join tree augmented with additional fields enables efficient computation of marginal probabilities of all variables fields: separators kept all potentials inside a clique attached to its node mailboxes Bayes Networks p. 13

14 Mihkli praktiline ülesanne Bayes Networks p. 14

15 Conclusion Calculating the marginal distributions is convenient via variable elimination the variable elimination sequence must end in the desired variable calcualting distributions for all variables separately isn t the most efficient way It is even more so using domain graphs representations If the domain graph is triangulated (has a perfect elimination sequence = no fill-ins) then There s a join tree for the domain set of the elimination sequence (graph cliques) All marginal distributions can be calculated efficiently via turing the join tree into a junction tree (regardless of the elimination sequence used to build the trees) Bayes Networks p. 15