Probabilistic Horn Abduction and Bayesian Networks

1. Probabilistic Horn abduction and Bayesian Networks David Poole presented by Hrishikesh Goradia Computer Science and Engineering, University of South Carolina

2. Introduction • Logic-based systems for diagnostic problems • Too many logical possibilities to handle • Many of the diagnoses not worth considering • Bayesian networks • Probabilistic analysis • Probabilistic Horn Abduction • Framework for logic-based abduction that incorporates probabilities with assumptions • Extends pure Prolog in a simple way to include probabilities Computer Science and Engineering, University of South Carolina

3. Motivating Example Computer Science and Engineering, University of South Carolina

4. Motivating Example Computer Science and Engineering, University of South Carolina

5. Probabilistic Horn Abduction Theory Computer Science and Engineering, University of South Carolina

6. Probabilistic Horn Abduction Theory Computer Science and Engineering, University of South Carolina

7. Assumptions and Constraints • Identical hypotheses cannot appear in multiple disjoint declarations. • All atoms in disjoint declarations share the same variables. • Hypotheses cannot form the head of rules. • No cycles in the knowledge base. • Knowledge base is both covering and disjoint. Computer Science and Engineering, University of South Carolina

8. Bayesian Networks to Probabilistic Horn Abduction Theory • A discrete Bayesian network is represented by Probabilistic Horn abduction rules that relates a random variable ai with its parents {ai1, …, ain}: • The conditional probabilities for the random variable are translated into assertions: Computer Science and Engineering, University of South Carolina

9. Bayesian Networks to Probabilistic Horn Abduction Theory Computer Science and Engineering, University of South Carolina

10. Bayesian Networks to Probabilistic Horn Abduction Theory Computer Science and Engineering, University of South Carolina

11. Probabilistic Horn Abduction Theory to Bayesian Networks • Each disjoint declaration maps to a random variable. • Each atom defined by rules also corresponds to a random variable. • Arcs go from the body RV(s) to the head RV in each rule. • Probabilities in the disjoint declarations map directly to the conditional probabilities for the RVs • Additional optimizations possible. Computer Science and Engineering, University of South Carolina

12. Discussion – Independence and Dependence • Can the world be represented such that all of the hypotheses are independent? Computer Science and Engineering, University of South Carolina

13. Discussion – Independence and Dependence • Can the world be represented such that all of the hypotheses are independent? • Author claims that it is possible. • Reichenbach’s principle of the common cause: “If coincidences of two events A and B occur more frequently than their independent occurrence, … then there exists a common cause for these events …” Computer Science and Engineering, University of South Carolina

14. Discussion – Abduction and Prediction • Is abducing to causes and making assumptions as to what to predict from those assumptions the right logical analogue of the independence in Bayesian networks? Computer Science and Engineering, University of South Carolina

15. Discussion – Abduction and Prediction • Is abducing to causes and making assumptions as to what to predict from those assumptions the right logical analogue of the independence in Bayesian networks? • Author claims that it is true. • Approach is analogous to Pearl’s network propagation scheme for computing conditional probabilities. Computer Science and Engineering, University of South Carolina

16. Discussion – Causation • Common problem associated with logical formulation of causation: “If c1is a cause for a and c2 is a cause for ¬a, then from c1 we can infer ¬c2.” Does the probabilistic Horn abduction theory overcome this? Computer Science and Engineering, University of South Carolina

17. Discussion – Causation • Common problem associated with logical formulation of causation: “If c1is a cause for a and c2 is a cause for ¬a, then from c1 we can infer ¬c2.” Does the probabilistic Horn abduction theory overcome this? • Author claims that it does. • The Bayesian network represented by the theory will have c1 and c2 as disjoint RVs. Computer Science and Engineering, University of South Carolina

18. Summary • Presents a simple framework for Horn clause abduction, with probabilities associated with hypotheses. • Finds a relationship between logical and probabilistic notions of evidential reasoning. • Presents a useful representation language that provides a compromise between heuristic and epistemic adequacy. Computer Science and Engineering, University of South Carolina