Probabilistic Horn abduction and Bayesian Networks David Poole presented by Hrishikesh Goradia 11/24/2018 Computer Science and Engineering, University of South Carolina

Computer Science and Engineering, University of South Carolina 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 11/24/2018 Computer Science and Engineering, University of South Carolina

Computer Science and Engineering, University of South Carolina Motivating Example 11/24/2018 Computer Science and Engineering, University of South Carolina

Computer Science and Engineering, University of South Carolina Motivating Example 11/24/2018 Computer Science and Engineering, University of South Carolina

Probabilistic Horn Abduction Theory 11/24/2018 Computer Science and Engineering, University of South Carolina

Probabilistic Horn Abduction Theory 11/24/2018 Computer Science and Engineering, University of South Carolina

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. 11/24/2018 Computer Science and Engineering, University of South Carolina

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: 11/24/2018 Computer Science and Engineering, University of South Carolina

Bayesian Networks to Probabilistic Horn Abduction Theory 11/24/2018 Computer Science and Engineering, University of South Carolina

Bayesian Networks to Probabilistic Horn Abduction Theory 11/24/2018 Computer Science and Engineering, University of South Carolina

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. 11/24/2018 Computer Science and Engineering, University of South Carolina

Discussion – Independence and Dependence Can the world be represented such that all of the hypotheses are independent? 11/24/2018 Computer Science and Engineering, University of South Carolina

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 …” 11/24/2018 Computer Science and Engineering, University of South Carolina

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? 11/24/2018 Computer Science and Engineering, University of South Carolina

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. 11/24/2018 Computer Science and Engineering, University of South Carolina

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? 11/24/2018 Computer Science and Engineering, University of South Carolina

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. 11/24/2018 Computer Science and Engineering, University of South Carolina

Computer Science and Engineering, University of South Carolina 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. 11/24/2018 Computer Science and Engineering, University of South Carolina