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Bayesian Abductive Logic Programs Sindhu Raghavan Raymond J. Mooney The University of Texas at Austin 1

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Abduction Process of finding the best explanation for a set of observations (Peirce 1958) Inference of cause from effect Applications – Plan recognition – Medical diagnosis – Natural language understanding 2

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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 (atomic facts) 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| 3

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Example - Plan Recognition Background knowledge B: go(person,loc) :- shopping(person,loc,item). go(person,loc) :- robbing(person,loc,instr). get(person,instr) :- robbing(person,loc,instr). get(person,instr) :- hunting(person,loc,instr). store(loc) :- shopping(person,loc,item). store(loc) :- robbing(person,loc,instr). gun(instr) :- robbing(person,loc,instr). gun(instr) :- hunting(person,loc,instr). 4

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Sample Observations O get(john,o1) gun(o1) go(john,p1) store(p1) 5

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Abductive Proof 1 get(john,o1) gun(o1) go(john,p1) store(p1) hunting(john,s2,o1)shopping(john,p1,s1) 6

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Abductive Proof 2 get(john,o1) gun(o1) go(john,p1) store(p1) robbing(john,p1,o1) 7

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Best Explanation 8 Explanation 1 hunting(john,s2,o1) shopping(john,p1,s1) Explanation 1 hunting(john,s2,o1) shopping(john,p1,s1) Explanation 2 robbing(john,p1,o1) Explanation 2 robbing(john,p1,o1) Best explanation makes fewest assumptions

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Existing Work on Logical Abduction History of research from 70’s – 90’s. (Pople 1973; Levesque 1989; Ng and Mooney 1992) Abductive Logic Programming (ALP) (Kakas, Kowalski, and Toni, 1993) – Formalization based on logic programming. 9

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Problem with Logical Abduction Not handle uncertainty of assumptions and inferences. Unable to chose between explanations with the same number of assumptions based on probability. 10

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Other Approaches to Abduction Probabilistic abduction using Bayesian networks (Pearl 1988). – Unable to capture relational structure since Bayes nets are propositional in nature Probabilistic relational abduction using Markov Logic Networks (MLNs) (Kate and Mooney 2009). – Does not use logical abduction, instead uses complex reverse implications to approximate it. 11

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Bayesian Logic Programs (BLP) (Kersting and De Raedt, 2001) Bayesian Logic Programs (BLPs) combine first- order logic and Bayesian networks. Deductive logic programming used to construct structure of a Bayes net. Not suitable for problems requiring abductive logical inference to form a proof structure by making assumptions. 12

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Bayesian Abductive Logic Programs (BALP) BLP ALP BALP Suitable for tasks involving abductive reasoning – plan recognition, diagnosis, etc. 13

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BLPs vs. BALPs Like BLPs, BALPs use logic programs as templates for constructing Bayesian networks. Unlike BLPs, BALPs uses logical abduction instead of deduction to construct the network. 14

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Abduction in BALPs Given : A set of observation literals O = {O 1, O 2,….O n } Compute all distinct abductive proofs of O. Construct a Bayesian network using the resulting set of proofs as in BLPs. Perform probabilistic inference on the Bayesian network to compute the best explanation. 15

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Abductive Proof 1 get(john,o1) gun(o1) go(john,p1) store(p1) hunting(john,s2,o1)shopping(john,p1,s1) 16

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Abductive Proof 2 get(john,o1) gun(o1) go(john,p1) store(p1) robbing(john,p1,o1) 17

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Resulting Bayes Net get(john,o1) gun(o1) go(john,p1) store(p1) hunting(john,s2,o1) shopping(john,p1,s1) robbing(john,p1,o1) 18

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Probabilistic Parameters As with BLPs, CPTs for Bayes net specified in first-order clauses. Noisy-and combining rule is used to specify the CPT for combining the conjuncts in the body of the clause – Reduces the number of parameters needed – Parameters can be learned from data Noisy-or combining rule is used to specify the CPT for combining the disjunctive contributions from different ground clauses with the same head – Models “explaining away” – Parameters can be learned from data 19

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Resulting Bayes Net get(john,o1) gun(o1) go(john,p1) store(p1) hunting(john,s2,o1) shopping(john,p1,s1) robbing(john,p1,o1) 20 noisy or

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Probabilistic Inference Specify truth value of observed facts. Compute the Most Probable Explanation (MPE) to determine the most likely combination of truth values to all unknown literals given this evidence. Use standard Bayes-net package E LVIRA for inference. 21

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Resulting Bayes Net get(john,o1) gun(o1) go(john,p1) store(p1) hunting(john,s2,o1) shopping(john,p1,s1) robbing(john,p1,o1) noisy or 22

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Resulting Bayes Net get(john,o1) gun(o1) go(john,p1) store(p1) hunting(john,s2,o1) shopping(john,p1,s1) robbing(john,p1,o1) Observed facts 23 noisy or

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Resulting Bayes Net get(john,o1) gun(o1) go(john,p1) store(p1) hunting(john,s2,o1) shopping(john,p1,s1) robbing(john,p1,o1) Observed facts Query variables 24 noisy or

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Resulting Bayes Net get(john,o1) gun(o1) go(john,p1) store(p1) hunting(john,s2,o1) shopping(john,p1,s1) robbing(john,p1,o1) Observed facts Query variables FALSE TRUE 25 noisy or

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Experimental Evaluation 26

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Story Understanding Data Recognizing plans from narrative text (Charniak and Goldman 1991; Ng and Mooney 1992). Infer characters’ higher-level plans that explain their observed actions represented in logic. – “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 25 development examples and 25 test examples 12.6 observations per example. Background knowledge base originally constructed for ACCEL system (Ng and Mooney 1992). 27

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Story Understanding Methodology Noisy-and and noisy-or parameters set to 0.9 and priors hand-tuned on development data. Multiple high-level plans per example are possible. MPE inference used to compute the best explanation. Computed precision, recall and F-measure. 28

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Story Understanding Systems Evaluated BALPs ACCEL – Simplicity (Ng and Mooney 1992) – Logical abduction preferring fewest # of assumptions. ACCEL – Coherence (Ng and Mooney 1992) – Logical abduction that maximally connects observations – Specific to story understanding. Abductive MLNs (Kate and Mooney 2009) 29

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Story Understanding Results 30

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Monroe Data Recognizing high level plans in an emergency response domain. Developed by Blaylock and Allen (2005) to test a statistical n-gram approach. 10 high level plans including setting up shelter, providing medical attention, clearing road wreck. Artificially generated using SHOP-2 HTN planner. 1000 examples for evaluation. 10.19 observation literals per example. Single correct plan in each example. Knowledge base constructed based on the domain knowledge encoded in HTN. 31

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Monroe Methodology Parameters were set as in story understanding. Computed marginal probabilities for all high level plans and selected the single one with the highest probability. Computed convergence score to compare with Blaylock and Allen results. Convergence score is the fraction of examples for which the top level plan schema (predicate only) was predicted accurately after seeing all observations. 32

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Monroe Results BALPBlaylock and Allen Convergence score 98.894.2 33

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Modified Monroe Data When applying the Kate & Mooney (2009) abductive MLN approach to Monroe, it resulted in explosively large ground networks. Simplified Monroe domain just enough to prevent these problems. Developed typed clauses effective for abductive MLNs. Weight learning was still not tractable, so weights set manually. 34

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Modified Monroe Methodology Repeatedly measured the percentage of correct plans inferred (including correct arguments) after observing an increasing fraction of the actions in the plan. 35

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Modified-Monroe Results % observations seen by the systems Accuracy 36

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Ongoing & Future Work Automatic learning of BALP parameters from data. Alternative MLN formulation more directly modeling the BALP approach. – Preliminary results for story understanding and modified Monroe are only slightly worse than BALP results, but much worse for original Monroe. Compare to other SRL approaches that have incorporated logical abduction (SLPs, P RISM ) when applied to plan recognition. Evaluation on other datasets/tasks/domains. 37

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Conclusions New SRL framework BALP that combines Bayesian Logic Programs and Abductive Logic Programming. Well suited for relational abductive reasoning tasks like plan recognition. Empirical results demonstrate advantages over existing methods. 38

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Questions?? 39

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