Probabilistic Abduction using Markov Logic Networks

1. Probabilistic Abduction using Markov Logic Networks Rohit J. Kate Raymond J. Mooney

2. Abduction • Abduction is inference to the best explanation for a given set of evidence • Applications include tasks in which observations need to be explained by the best hypothesis • Plan recognition • Intent recognition • Medical diagnosis • Fault diagnosis … • Most previous work falls under two frameworks for abduction • First-order logic based Abduction • Probabilistic abduction using Bayesian networks

3. Logical Abduction Given: • Background knowledge, B, in the form of a set of (Horn) clauses in first-order logic • Observations, O, in the form of atomic facts in first-order logic Find: • A hypothesis, H, a set of assumptions (logical formulae) that logically entail the observations given the theory BH  O • Typically, best explanation is the one with the fewest assumptions, e.g. minimizes |H|

4. Sample Logical Abduction Problem • Background Knowledge: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria)) • Observations: Infected(John, Malaria) Transfuse(Blood, Mary, John) • Explanation: Infected(Mary, Malaria)

5. Previous Work in Logical Abduction • Several first-order logic based approaches [Poole et al. 1987; Stickel 1988; Ng & Mooney 1991; Kakas et al. 1993] • Perform first-order “backward” logical reasoning to determine the set of assumptions sufficient to deduce observations • Unable to reason under uncertaintyto find the most probable explanation Background Knowledge: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) Holds 80% of the times x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria)) Holds 40% of the times Observation: Infected(John, Malaria) 99% sure Transfuse(Blood, Mary, John) 60% sure

6. Previous Work in Probabilistic Abduction • An alternate framework is based on Bayesian networks [Pearl 1988] • Uncertainties are encoded in a directed graph • Given a set of observations, probabilistic inference over the graph computes the posterior probabilities of explanations • Unable to handle structured representations because essentially based on propositional logic

7. Probabilistic Abduction using MLNs • We present a new approach for probabilistic abduction that combines first-order logic and probabilistic graphical models • Uses Markov Logic Networks (MLNs)[Richardson and Domingos 2006], a theoretically sound framework for combining first-order logic and probabilistic graphical models Rest of the talk: • MLNs • Our approach using MLNs • Experiments • Future Work and Conclusions

8. Markov Logic Networks (MLNs) [Richardson and Domingos 2006] • A logical knowledge base is a set of hard constraintson the set of possible worlds • An MLN is a set of soft constraints:When a world violates a clause, it becomes less probable, not impossible • Give each clause a weight(Higher weight  Stronger constraint)

9. Sample MLN Clauses x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) 20 x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria)) 5

10. MLN Probabilistic Model MLN is a template for constructing a Markov network Ground literals correspond to nodes Ground clauses correspond to cliques connecting the ground literals in the clause Probability of a world (truth assignments) x: Weight of clause i No. of true groundings of clause i in x

11. Sample MLN Probabilistic Model • Clauses with weights: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria))20 x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria))5 • Constants: John, Mary, M • Ground literals: Mosquito(M) Infected(M,Malaria) Bite(M,John) Bite(M,Mary) Infected(John,Malaria) Infected(Mary,Malaria) Transfuse(Blood,John,Mary) Transfuse(Blood,Mary,John)

12. Sample MLN Probabilistic Model • Clauses with weights: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria))20 x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria))5 • Constants: John, Mary, M • Ground literals: Mosquito(M) true Infected(M,Malaria) true Bite(M,John) false Bite(M,Mary) false Infected(John,Malaria) true Infected(Mary,Malaria) true Transfuse(Blood,John,Mary) true Transfuse(Blood,Mary,John) false

13. Sample MLN Probabilistic Model • Clauses with weights: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) 20  x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria))5 • Constants: John, Mary, M • Ground literals: Mosquito(M) true Infected(M,Malaria) true Bite(M,John) false Bite(M,Mary) false Infected(John,Malaria) true Infected(Mary,Malaria) true Transfuse(Blood,John,Mary) true Transfuse(Blood,Mary,John) false

14. Sample MLN Probabilistic Model • Clauses with weights: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) 20   x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria))5 • Constants: John, Mary, M • Ground literals: Mosquito(M) true Infected(M,Malaria) true Bite(M,John) false Bite(M,Mary) false Infected(John,Malaria) true Infected(Mary,Malaria) true Transfuse(Blood,John,Mary) true Transfuse(Blood,Mary,John) false

15. Sample MLN Probabilistic Model • Clauses with weights: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria))20   x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria)) 5  • Constants: John, Mary, M • Ground literals: Mosquito(M) true Infected(M,Malaria) true Bite(M,John) false Bite(M,Mary) false Infected(John,Malaria) true Infected(Mary,Malaria) true Transfuse(Blood,John,Mary) true Transfuse(Blood,Mary,John) false

16. Sample MLN Probabilistic Model • Clauses with weights: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria))20   x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria)) 5   • Constants: John, Mary, M • Ground literals: Mosquito(M) true Infected(M,Malaria) true Bite(M,John) false Bite(M,Mary) false Infected(John,Malaria) true Infected(Mary,Malaria) true Transfuse(Blood,John,Mary) true Transfuse(Blood,Mary,John) false

17. Sample MLN Probabilistic Model • Clauses with weights: x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria))20   x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria))5   • Constants: John, Mary, M • Ground literals: Mosquito(M) true Infected(M,Malaria) true Bite(M,John) false Bite(M,Mary) false Infected(John,Malaria) true Infected(Mary,Malaria) true Transfuse(Blood,John,Mary) true Transfuse(Blood,Mary,John) false

18. MLNs Inference and Learning • Using probabilistic inference techniques one can determine the most probable truth assignment, probability that a clause holds etc. • Given a database of training examples, appropriate weights of the formulae can be learned to maximize the probability of the training data • An open-source software package for MLNs called Alchemy is available

19. Abduction using MLNs • Given: Infected(Mary,Malaria)  Transfuse(Blood,Mary,John) → Infected(John,Malaria)) Transfuse(Blood, Mary, John) Infected(John, Malaria) • The clause is satisfied whether Infected(Mary, Malaria) is true or false • Given the observations, a world has the same probability in MLN whether the explanation is true or false, explanations cannot be inferred • The MLN inference mechanism is inherently deductive and not abductive

20. Adapting MLNs for Abduction • Explicitly include the reverse implications x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria)) y (Infected(y,Malaria) → x (Transfuse(Blood,x,y)  Infected(x,Malaria))) • Existentially quantify the universally quantified variables which appear on the LHS but not on the RHS in the original clause • Now, given Transfuse(Blood, Mary, John) and Infected(John, Malaria),the probability of the world in which Infected(Mary,Malaria) is true will be higher

21. Adapting MLNs for Abduction • However, there could be multiple explanations for the same observations: x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria)) y (Infected(y,Malaria) → x (Transfuse(Blood,x,y)  Infected(x,Malaria))) x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) y (Infected(y,Malaria) → x (Mosquito(x)  Infected(x,Malaria)  Bite(x,y))) • An observation should be explained by one explanation and not multiple explanations • The system should support “explaining away” [Pearl 1988]

22. Adapting MLNs for Abduction x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) x y (Infected(x,Malaria)  Transfuse(Blood,x,y) → Infected(y,Malaria)) • Add the disjunction clause and the mutual exclusivity clause for the same RHS term • Since MLN clauses are soft constraints both explanations can still be true • y (Infected(y,Malaria) → • x (Transfuse(Blood,x,y)  Infected(x,Malaria))) v • x (Mosquito(x)  Infected(x,Malaria)  Bite(x,y))) • y (Infected(y,Malaria) → • ( x (Transfuse(Blood,x,y)  Infected(x,Malaria))) v • (x (Mosquito(x)  Infected(x,Malaria)  Bite(x,y))))

23. Adapting MLNs for Abduction • In general, for the Horn clauses P1→ Q,P2→ Q , …, Pn→ Q in the background knowledge base, add: • A reverse implication disjunction clause Q → P1 v P2v… v Pn • A mutual exclusivity clause for every pair of explanations Q →  P1 v P2 Q →  P1 v Pn … Q →  P2 v Pn • Weights can be learned from training examples or can be set heuristically

24. Adapting MLNs for Abduction • There could be constants or variables on the RHS predicate x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) x (Infected(x,Malaria)  Transfuse(Blood,x,John) → Infected(John,Malaria))

25. Adapting MLNs for Abduction • There could be constants or variables on the RHS predicate x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) x (Infected(x,Malaria)  Transfuse(Blood,x,John) → Infected(John,Malaria)) • Infected(John,Malaria) → • x (Transfuse(Blood,x,John)  Infected(x,Malaria))) v • x (Mosquito(x)  Infected(x,Malaria)  Bite(x,John)) • Infected(John,Malaria) → • ( x (Transfuse(Blood,x,John)  Infected(x,Malaria))) v • (x (Mosquito(x)  Infected(x,Malaria)  Bite(x,John)))

26. Adapting MLNs for Abduction • There could be constants or variables on the RHS predicate x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) x (Infected(x,Malaria)  Transfuse(Blood,x,John) → Infected(John,Malaria)) • Infected(John,Malaria) → • x (Transfuse(Blood,x,John)  Infected(x,Malaria))) v • x (Mosquito(x)  Infected(x,Malaria)  Bite(x,John)) • Infected(John,Malaria) → • ( x (Transfuse(Blood,x,John)  Infected(x,Malaria))) v • (x (Mosquito(x)  Infected(x,Malaria)  Bite(x,John))) • y (Infected(y,Malaria) → • x (Mosquito(x)  Infected(x,Malaria)  Bite(x,y)))

27. Adapting MLNs for Abduction • There could be constants or variables on the RHS predicate x y (Mosquito(x)  Infected(x,Malaria)  Bite(x,y) → Infected(y,Malaria)) x (Infected(x,Malaria)  Transfuse(Blood,x,John) → Infected(John,Malaria)) • Infected(John,Malaria) → • x (Transfuse(Blood,x,John)  Infected(x,Malaria))) v • x (Mosquito(x)  Infected(x,Malaria)  Bite(x,John)) • Infected(John,Malaria) → • ( x (Transfuse(Blood,x,John)  Infected(x,Malaria))) v • (x (Mosquito(x)  Infected(x,Malaria)  Bite(x,John))) • y (Infected(y,Malaria) → • x (Mosquito(x)  Infected(x,Malaria)  Bite(x,y))) • Formal algorithm is described in the paper, requires appropriate unifications and variable re-namings

28. Experiments: Dataset • Plan recognition dataset used to evaluate abductive systems [Ng & Mooney 1991; Charniak & Goldman 1991] • Character’s higher-level plans must be inferred to explain their observed actions in a narrative text • “Fred went to the supermarket. He pointed a gun at the owner. He packed his bag.” => robbing • “Jack went to the supermarket. He found some milk on the shelf. He paid for it.” => shopping • Dataset contains 25 development [Goldman 1990] and 25 test examples [Ng & Mooney 1992]

29. Experiments: Dataset contd. • Background knowledge-base was constructed for the ACCEL system [Ng and Mooney 1991] to work with the 25 development examples; 107 such rules instance_shopping(s) ^ go_step(s,g)  instance_going(g) instance_shopping(s) ^ go_step(s,g) ^ shopper(s,p)  goer(g,p) • Narrative text is represented in first order logic; average 12.6 literals • “Bill went to the store. He paid for some milk.” instance_going(Go1) goer(Go1,Bill) destination_go(Store1) instance_paying(Pay1) payer(Pay1,Bill) thing_paid(Pay1,Milk1) • Assumptions explaining the above actions instance_shopping(S1) shopper(S1,Bill) go_step(S1,Go1) pay_step(S1,Pay1) thing_shopped_for(S1,Milk)

30. Experiments: Methodology • Our algorithm automatically adds clauses to the knowledge-base for performing abduction using MLNs • We found that 25 development examples were too few to learn weights for MLNs, we heuristically set the weights • Small negative weights on unit clauses so that they are not assumed for no reason • Medium weights on reverse implication clauses • Large weights on mutual exlcusivity clauses • Given a set of observations, we use Alchemy’s probabilistic inference to determine the most likely truth assignment for the remaining literals

31. Experiments: Methodology contd. • We compare with the ACCEL system [Ng & Mooney 1992], a purely logic-based system for abduction • Selects the best explanation using a metric • Simplicity metric: selects the explanation of smallest size • Coherence metric: selects the explanation that maximally connects the observations (specifically geared towards this task) • “John took the bus. He bought milk.” => John took the bus to the store where he bought the milk.

32. Experiments: Methodology contd. • Besides finding the assumptions, a deductive system like MLN also finds other facts that can be deduced from the assumptions • We deductively expand ACCEL’s output and gold-standard answers for a fair comparison • We measure • Precision: what fraction of the predicted ground literals are in the gold-standard answers • Recall: what fraction of the ground literals in the gold-standard answers were predicted • F-measure: harmonic mean of precision and recall

33. Experiments: Results Development Set

34. Experiments: Results contd. Test Set

35. Experiments: Results contd. • MLN performs better than ACCEL-simplicity particularly on the development set • ACCEL-coherence performs the best, but was specifically tailored for narrative understanding task • The dataset used does not require full probabilistic treatment because little uncertainty in the knowledge-base or observations • MLNs did not need any heuristic metric but simply found the most probable explanation

36. Future Work • Evaluate probabilistic abduction using MLNs on a task in which uncertainty plays a bigger role • Evaluate on a larger dataset on which the weights could be learned to automatically adapt to a particular domain • Previous abductive systems like ACCEL have no learning mechanism • Perform probabilistic abduction using other frameworks of combining first-order logic and graphical models [Getoor & Taskar 2007], for example, Bayesian Logic Programming [Kersting & De Raedt 2001] and compare with the presented approach

37. Conclusions • A general method for probabilistic first-order logical abduction using MLNs • Existing off-the-shelf deductive inference system of MLNs is employed to do abduction by suitably reversing the implications • Handles uncertainties using probabilties and an unbounded number of related entities using first-order logic, capable of learning • Experiments on a small plan recognition dataset demonstrated that it compares favorably with special-purpose logic-based abductive systems

38. Thanks! Questions?