Bayesian Networks: Review Representation, independence, and inference (+ a few problems for your enjoyment)

1. Bayesian Networks: ReviewRepresentation, independence, and inference(+ a few problems for your enjoyment) Stanislav Funiak 10-701 Recitation, 3/30/2006 Bayesian Networks: Review

2. Conference Submission Network Beer Done in Time Sleep1 Quality Sleep2 Comm- ents1 Comm-ents2 Recom-mended Accepted Bayesian Networks: Review

3. B D T S1 Q S2 C1 C2 R A Bayesian Network • DAG, each node is a variable • For each var. X, CPD p(X | Pa X) • represents the distribution as A,B,D,T,Q,S1,S2,C1,C2,R Bayesian Networks: Review

4. S1 Q C1 C2 R A Smaller Network Q=t C2=t Q=f C2=f p(S1=f, Q=t, C1=f, C2=t, R=t, A=t) = Bayesian Networks: Review

5. S1 Q C1 C2 R A Independence Relations • … where the force lies • example independencies Bayesian Networks: Review

6. Factorization => Independence relations • We have seen: • starting from: p factorizes according to G • show: p satisfies some independence relations • Which independence assumptions in G? p factorizes according to G p satisfies indep. relations in G Bayesian Networks: Review

7. B D T S1 Q S2 C1 C2 R A Independence relations encoded in G 1. Local Markov Assumptions: • X indep. NonDescendants(X) | Pa X Bayesian Networks: Review

8. B D T S1 Q S2 C1 C2 R A Independence relations encoded in G 2. absence of active trails • Variables X indep of variables Ygiven Z if no active trail betweenX and Y given Z Bayesian Networks: Review

9. Active trail: Review • A path X1 –X2 –· · · –Xk is an active trail when variables Oµ{X1,…,Xn} are observed if for each consecutive triplet in the trail: • Xi-1XiXi+1,and Xi is not observed (XiO) • Xi-1XiXi+1,and Xi is not observed (XiO) • Xi-1XiXi+1,and Xi is not observed (XiO) • Xi-1XiXi+1,and Xiis observed (Xi2O), or one of its descendents Bayesian Networks: Review

10. B D T S1 Q S2 C1 C2 R A Independence relations encoded in G 2. absence of active trails • Variables X indep of variables Ygiven Z if no active trail betweenX and Y given Z Bayesian Networks: Review

11. Independence relations encoded in G 1. Local Markov Assumptions 2. absence of active trails (d-separation) Bayesian Networks: Review

12. Independence relations => Factorization • Before: • How about: p factorizes according to G p satisfies indep. relations in G p factorizes according to G p satisfies indep. relations in G Bayesian Networks: Review

13. S1 Q C1 C2 R A Independence relations => Factorization • Suppose • Prove that p factorizes according to G Bayesian Networks: Review

14. Queries • Marginal probability • Most probable explanation (MPE) • Active data collection Variable elimination Bayesian Networks: Review

15. Conditioning on evidence S1 Q C1 C2 R Q=t C2=t A Q=f C2=f Bayesian Networks: Review

16. Marginal probability query • Let’s compute S1 Q C1 C2 R A Bayesian Networks: Review

17. Most probable explanation • Now, let’s compute S1 Q C1 C2 R A Bayesian Networks: Review

18. Do we need this algorithm? • Couldn’t we just take argmax of marginals? Take Bayesian Networks: Review

19. What you need to know • Representation • Independence relations • local Markov assumption • active trails / d-separation • independence relations => factorization • Variable elimination • marginal queries • argmax queries Bayesian Networks: Review