Dynamic Probabilistic Relational Models Paper by: Sumit Sanghai, Pedro Domingos, Daniel Weld Anna Yershova Presentation slides are adapted from: Lise Getoor, Eyal Amir and Pedro Domingos slides

The problem How to represent/model uncertain sequential phenomena?

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.

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

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

Particle Filtering

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.

Particle Filter Algorithm Create particles as samples from the initial state distribution p(A 1, B 1, C 1 ). 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.

Initial State Distribution A 1, B 1, C 1

Prediction A t , B t , C t  A t, B t, C t = f (A t , B t , C t  ) A t, B t, C t

Compute Weights Before After A t, B t, C t

Resample A t, B t, C t

Rao-Blackwellised PF

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

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 Part2 1 - p f Uniform distr. over the rest of the objects true P(Part1.mate | Bolt(Part1, Part2))

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

Fault Model Used With probability 1  p f an action produces the intended effect, with probability p f one of the several faults occur: Painting not being completed Wrong color used Bolting the wrong object Welding the wrong object

Observation Model Used With probability 1  p o the truth value of the attribute is observed, with probability p o an incorrect value is observed

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

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)

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.

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