BLOG: Probabilistic Models with Unknown Objects

1. BLOG: Probabilistic Models with Unknown Objects Brian Milch, Bhaskara Marthi, Stuart Russell,David Sontag, Daniel L. Ong, Andrey Kolobov University of California at Berkeley

2. Basic Task • Given observations, make inferences about underlying objects • Difficulties: • Don’t know list of objects in advance • Don’t know when same object observed twice (identity uncertainty / data association / record linkage)

3. Handling Unknown Objects • Standard practice: special-purpose algorithms to resolve identity uncertainty • Goal: Resolve identity uncertainty by inference in probabilistic model • Bayesian LOGic (BLOG): representation language for models with • Unknown set of objects • Unknown map from observations to objects

4. Outline • Motivating applications • Bayesian Logic (BLOG) • Syntax • Semantics • Proof-of-concept experimental results

5. Example 1: Aircraft Tracking DetectionFailure

6. Example 1: Aircraft Tracking UnobservedObject FalseDetection

7. Russell, Stuart and Norvig, Peter. Articial Intelligence. Prentice-Hall, 1995. S. Russel and P. Norvig (1995). Artificial Intelligence: A Modern Approach. Upper Saddle River, NJ: Prentice Hall. Example 2: Bibliographies

8. Simple Example: Balls in an Urn P(n balls in urn) P(n balls in urn | draws) Draws (with replacement) 1 2 3 4

9. Draws Draws Draws Draws Draws Possible Worlds 3.00 x 10-3 7.61 x 10-4 1.19 x 10-5 … … 2.86 x 10-4 1.14 x 10-12 … …

10. Distributions over First-Order Structures • Idea goes back to Gaifman [1964] • Halpern [1990] defines language for stating constraints on such distributions • But not specifying a distribution uniquely • Logic programming approaches [Poole 1993; Sato & Kameya 2001; Kersting & De Raedt 2001] define unique distributions, but assume unique names and domain closure • PRMs [Koller & Pfeffer 1998] have special constructs for number uncertainty, existence uncertainty • BLOG: Unified syntax for distributions over worlds with: • Varying sets of objects • Varying mappings from observations to objects See also MEBN [Laskey and da Costa, UAI 2005]

11. Generative Process for Possible Worlds Draws (with replacement) 1 2 3 4

12. BLOG Model for Urn and Balls type Color; type Ball; type Draw; random Color TrueColor(Ball);random Ball BallDrawn(Draw);random Color ObsColor(Draw); guaranteed Color Blue, Green;guaranteed Draw Draw1, Draw2, Draw3, Draw4; #Ball ~ Poisson[6](); TrueColor(b) ~ TabularCPD[[0.5, 0.5]](); BallDrawn(d) ~ UniformChoice({Ball b}); ObsColor(d) if (BallDrawn(d) != null) then ~ NoisyCopy(TrueColor(BallDrawn(d)));

13. BLOG Model for Urn and Balls type Color; type Ball; type Draw; random Color TrueColor(Ball);random Ball BallDrawn(Draw);random Color ObsColor(Draw); guaranteed Color Blue, Green;guaranteed Draw Draw1, Draw2, Draw3, Draw4; #Ball ~ Poisson[6](); TrueColor(b) ~ TabularCPD[[0.5, 0.5]](); BallDrawn(d) ~ UniformChoice({Ball b}); ObsColor(d) if (BallDrawn(d) != null) then ~ NoisyCopy(TrueColor(BallDrawn(d))); header number statement dependencystatements

14. BLOG Model for Urn and Balls type Color; type Ball; type Draw; random Color TrueColor(Ball);random Ball BallDrawn(Draw);random Color ObsColor(Draw); guaranteed Color Blue, Green;guaranteed Draw Draw1, Draw2, Draw3, Draw4; #Ball ~ Poisson[6](); TrueColor(b) ~ TabularCPD[[0.5, 0.5]](); BallDrawn(d) ~ UniformChoice({Ball b}); ObsColor(d) if (BallDrawn(d) != null) then ~ NoisyCopy(TrueColor(BallDrawn(d))); ? Identity uncertainty: BallDrawn(Draw1) = BallDrawn(Draw2)

15. BLOG Model for Urn and Balls type Color; type Ball; type Draw; random Color TrueColor(Ball);random Ball BallDrawn(Draw);random Color ObsColor(Draw); guaranteed Color Blue, Green;guaranteed Draw Draw1, Draw2, Draw3, Draw4; #Ball ~ Poisson[6](); TrueColor(b) ~ TabularCPD[[0.5, 0.5]](); BallDrawn(d) ~ UniformChoice({Ball b}); ObsColor(d) if (BallDrawn(d) != null) then ~ NoisyCopy(TrueColor(BallDrawn(d))); Arbitrary conditionalprobability distributions CPD arguments

16. BLOG Model for Urn and Balls type Color; type Ball; type Draw; random Color TrueColor(Ball);random Ball BallDrawn(Draw);random Color ObsColor(Draw); guaranteed Color Blue, Green;guaranteed Draw Draw1, Draw2, Draw3, Draw4; #Ball ~ Poisson[6](); TrueColor(b) ~ TabularCPD[[0.5, 0.5]](); BallDrawn(d) ~ UniformChoice({Ball b}); ObsColor(d) if (BallDrawn(d) != null) then ~ NoisyCopy(TrueColor(BallDrawn(d))); Context-specific dependence

17. BLOG Model for Urn and Balls type Color; type Ball; type Draw; random Color TrueColor(Ball);random Ball BallDrawn(Draw);random Color ObsColor(Draw); guaranteed Color Blue, Green;guaranteed Draw Draw1, Draw2, Draw3, Draw4; #Ball ~ Poisson[6](); TrueColor(b) ~ TabularCPD[[0.5, 0.5]](); BallDrawn(d) ~ UniformChoice({Ball b}); ObsColor(d) if (BallDrawn(d) != null) then ~ NoisyCopy(TrueColor(BallDrawn(d)));

18. Generative Process for Aircraft Tracking Existence of radar blips depends on existence and locations of aircraft Sky Radar

19. BLOG Model for Aircraft Tracking Source … #Aircraft ~ NumAircraftDistrib(); State(a, t) if t = 0 then ~ InitState() else ~ StateTransition(State(a, Pred(t))); #Blip: (Source, Time) -> (a, t) ~ NumDetectionsDistrib(State(a, t)); #Blip: (Time) -> (t) ~ NumFalseAlarmsDistrib(); ApparentPos(r)if (Source(r) = null) then ~ FalseAlarmDistrib()else ~ ObsDistrib(State(Source(r), Time(r))); a Blips t 2 Time Blips t 2 Time

20. Declarative Semantics • What is the set of possible worlds? • What is the probability distribution over worlds?

21. What Exactly Are the Objects? • Objects are tuples that encode generation history • Aircraft: (Aircraft, 1),(Aircraft, 2), … • Blip from (Aircraft, 2) at time 8:(Blip, (Source, (Aircraft, 2)), (Time, 8), 1)

22. Basic Random Variables (RVs) • For each number statement and tuple of generating objects, have RV for number of objects generated • For each function symbol and tuple of arguments, have RV for function value • Lemma: Full instantiation of these RVs uniquely identifies a possible world

23. Another Look at a BLOG Model … #Ball ~ Poisson[6](); TrueColor(b) ~ TabularCPD[[0.5, 0.5]](); BallDrawn(d) ~ UniformChoice({Ball b}); ObsColor(d) if !(BallDrawn(d) = null) then ~ NoisyCopy(TrueColor(BallDrawn(d))); Dependency and number statements define CPDs for basic RVs

24. Just a Bayes Net? #Ball TrueColor(B1) TrueColor(B2) TrueColor(B3) … ObsColor(D1) Infinite parent set BallDrawn(D1) Standard BN results no longer apply

25. Probability Distribution • BLOG model specifies: • Conditional distributions for basic RVs • Factorization properties for certain finite instantiations of basic RVs • Theorem: Under certain conditions (analogous to BN acyclicity), every BLOG model defines unique distribution over possible worlds

26. Inference • Does infinite set of basic RVs prevent inference? • No: Sampling algorithm only needs to instantiate finite set of relevant variables • Algorithms: • Rejection sampling [this paper] • Guided likelihood weighting [Milch et al., AI/Stats 2005] • Theorem: For large class of BLOG models, sampling algorithms converge to correct probability for any query, using finite time per sampling step

27. Proof-Of-Concept Experiment • Given 10 draws, all appearing blue • 5 runs of 100,000 samples each prior posterior

28. Conclusions • Bayesian logic (BLOG) models define unique distributions over first-order model structures with • Varying sets of objects • Varying mappings from terms to objects • Future work: • Practical inference algorithms • Applications to text understanding • Applications to situation awareness (DBLOG)