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 Detection Failure

6. Example 1: Aircraft Tracking False Detection Unobserved Object

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

8. Simple Example: Balls in an Urn Draws (with replacement) P(n balls in urn) P(n balls in urn | draws) 1234

9. 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 Draws

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) 1234

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 dependency statements

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 conditional probability 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 SkyRadar Existence of radar blips depends on existence and locations of aircraft

19. BLOG Model for Aircraft Tracking … #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))); 2 Source Time a t Blips 2 Time t Blips

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? TrueColor(B1 ) TrueColor(B2 ) TrueColor(B3 ) … ObsColor(D1 ) BallDrawn(D1 ) #Ball Infinite parent set 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 prior posterior Given 10 draws, all appearing blue 5 runs of 100,000 samples each

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)