Dynamic Probabilistic Relational Models Paper by: Sumit Sanghai, Pedro Domingos, Daniel Weld

1. Dynamic Probabilistic Relational ModelsPaper by: Sumit Sanghai, Pedro Domingos, Daniel Weld Anna Yershova yershova@uiuc.edu Presentation slides are adapted from: Lise Getoor, Eyal Amir and Pedro Domingos slides

2. The problem How to represent/model uncertain sequential phenomena?

4. Limitations of the DBNs How to represent: • Classes of objects and multiple instances of a class • Multiple kinds of relations • Relations evolving over time Example: Early fault detection in manufacturing Complex and diverse relations evolving over the manufacturing process.

5. BRACKET weight PLATE shape PLATE PLATE PLATE PLATE PLATE PLATE PLATE PLATE PLATE PLATE PLATE weight color shape weight weight weight weight weight weight weight weight weight weight weight welded to color shape shape shape shape shape shape shape shape shape shape shape bolt welded to color color color color color color color color color color color bolt BOLT weight size type welded to bolt Fault detection in manufacturing ACTION action

10. Strain Strain Patient Patient Contact Contact Objects Attributes DPRM with AU Semantics • Contact • c1 • Contact • c1 Strain s1 Strain s1 Patient p2 • Contact • c2 Patient p2 Strain s2 • Contact • c2 Strain s2 Patient p1 • Contact • c3 Patient p1 • Contact • c3 Patient p3 Patient p3 relational skeletons 1,2 2TPRM + = probability distribution over completions I:

12. Particle Filtering

13. The Objective of PF • The objective of the particle filter is to compute the conditional distribution • To do this analytically - expensive • The particle filter gives us an approximate computational technique.

14. Particle Filter Algorithm • Create particles as samples from the initial state distribution p(A1, B1, C1). • For i going from 1 to N • Update each particle using the state update equation. • Compute weights for each particle using the observation value. • (Optionally) resample particles.

15. Initial State Distribution A1, B1, C1 A1, B1, C1

16. Prediction At-1, Bt-1, Ct-1 At, Bt, Ct = f (At-1, Bt-1, Ct-1 ) At, Bt, Ct

17. Compute Weights At, Bt, Ct Before At, Bt, Ct After At, Bt, Ct

18. Resample At, Bt, Ct At, Bt, Ct

21. Rao-Blackwellised PF

23. Another Issue • Rao-Blackwellising the relational attributes can vastly reduce the size of the state space. • If the relational skeleton contains a large number of objects and relations, storing and updating all the requisite probabilities can still become quite expensive. • Use some particular knowledge of the domain

24. Abstraction trees • Replace the vector of probabilities with a tree structure • leaves represent probabilities for entire sets of objects • nodes represent all combinations of the propositional attributes true P(Part1.mate | Bolt(Part1, Part2)) Uniform distr. over the rest of the objects Part2 1 - pf

25. BRACKET weight PLATE shape PLATE PLATE PLATE PLATE PLATE PLATE PLATE PLATE PLATE PLATE PLATE weight color shape weight weight weight weight weight weight weight weight weight weight weight welded to color shape shape shape shape shape shape shape shape shape shape shape bolt welded to color color color color color color color color color color color bolt BOLT weight size type welded to bolt Experiments Dom(ACTION.action) = {paint, drill, polish, Change prop. Attr., Change rel. Attr.} ACTION action

26. Fault Model Used With probability 1 -pf an action produces the intended effect, with probability pf one of the several faults occur: • Painting not being completed • Wrong color used • Bolting the wrong object • Welding the wrong object

27. Observation Model Used With probability 1 -po the truth value of the attribute is observed, with probability po an incorrect value is observed

28. Measure of the Accuracy • K-L divergence between distributions • Computing is infeasible – approximation is needed

29. Approximation of K-L Divergence • We are interested only in measuring the differences in performance of different approximation methods -> first term is eliminated • Take S samples from the true distribution (S = 10,000 in the experiments)

32. Experimental Results • Abstraction trees reduced RBPF’s time and memory by a factor of 30 to 70 • On average six times longer and 11 times the memory of PF, per particle. • However, note that we ran PF with 40 times more particles than RBPF. • Thus, RBPF is using less time and memory than PF, and performing far better in accuracy.

33. Conclusions and future work • Relaxing the assumptions made • Further scaling up inference • Studying the properties of the abstraction trees • Handling continuous variables • Learning DPRMs • Applying them to the real world problems