Dynamic Probabilistic Relational Models

Paper 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

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.

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Fault detection in manufacturing

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BRACKETweightshapecolor

welded tobolt

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BOLTweightsizetype

welded tobolt

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welded tobolt

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ACTION

action

Strain s1

Patient p1

Patient p2

Contactc3

Contactc2

Contactc1

Strain s2

Patient p3

Strain

Patient

Contact

DPRM with AU Semantics

)).(|.(),S,|( ,.

AxparentsAxPP Sx Ax

I

AttributesObjects

probability distribution over completions I:

2TPRM relational skeletons 12+ =

Strain

Patient

Contact

Strain s1

Patient p1

Patient p2

Contactc3

Contactc2

Contactc1

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(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.

Initial State Distribution

A1, B1, C1

A1, B1, C1

Prediction

At, Bt, Ct

At, Bt, Ct = f (At, Bt, Ct )

At, Bt, Ct

Compute Weights

Before

After

At, Bt, Ct

At, Bt, Ct

At, Bt, Ct

Resample

At, Bt, Ct

At, Bt, Ct

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

Part21 - pf

Uniform distr. over the rest of the objects

trueP(Part1.mate | Bolt(Part1, Part2))

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Experiments

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BRACKETweightshapecolor

welded tobolt

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BOLTweightsizetype

welded tobolt

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ACTION

action

Dom(ACTION.action) ={paint, drill, polish,Change prop. Attr.,Change rel. Attr.}

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

Observation Model Used

With probability 1 po the truth value of the attribute is observed, with probability po 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