toothache toothache

catch catch catch catch

cavity 0.108 0.012 0.072 0.008

cavity 0.016 0.064 0.144 0.576

Joint PDF

Structure and Semantics of BN• draw causal nodes first• draw directed edges to effects (“direct causes”)• links encode conditional probability tables

(CPT over parents)• fewer parameters than full joint PDF• absence of link is related to independence

• child is cond.dep. on parent: P(B|A)

• parent is cond.dep. on child:– P(A|B)=P(B|A)P(A)/P(B)

• what about when one node is not an ancestor of the other? e.g. siblings

A

B

A and B are only conditionally independent given C

simple treespoly-trees (singly connected, one path between any pair of nodes)“cyclic” (using undirected edges) – much harder to do computations

explaining away: P(sprinkler | wetGrass) = 0.43P(sprinkler | wetGrass,rain) = 0.19

A Bayesian network approach to threat valuation with application to an air defense scenario, Johansson and Falkman

Lumiere – Office Assistant

Inference Tasks• posterior: P(Xi|{Zi})

– Zi observed vars, with unobserved variables Yi, marginalized out– prediction vs. diagnosis– evidence combination is crucial– handling unobserved variables is crucial

• all marginals: P(Ai) – like priors, but for interior nodes too• subjoint: P(A,B)• boolean queries• most-probable explanation:

– argmax{Yi} P(Yi U Zi) – state with highest joint probability

(see slides 4-10 in http://aima.eecs.berkeley.edu/slides-pdf/chapter14b.pdffor discussion of Enumeration and VariableElimination)

from: Inference in Bayesian Networks, D’Ambrosio

full joint PDF:

sub-joint

conditional (normalized):

Belief Propagation (this figure happens to come from http://www.pr-owl.org/basics/bn.php)see also: wiki, Ch. 8 in Bishop PR&ML