l19bwintroprob

l19bwintroprob - 3/6/00 Introduction to Probabilistic...

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3/6/00 1 9/13/00 copyright Brian Wil iams, 2000 1 courtesy of JPL Introduction to Probabilistic Reasoning Brian C. Williams 16.410/16.413 Session 19 Brian C. Wil iams, copyright 2000 9/13/00 copyright Brian Wil iams, 2000 2 Reading Assignments AIMA (Russell and Norvig) § Ch 13 Review of Probabilities § Ch 14.1-4 Probabilistic Reasoning § Ch 15.1-.3, 20.3 State Estimation and Hidden Markov Models § No homework due this week. 9/13/00 copyright Brian Wil iams, 2000 3 Outline § Motivation § Set Theoretic View of Propositional Logic § From Propositional Logic to Probabilities § Probabilistic Inference Methods § Appendix: Application to Model-based Diagnosis
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3/6/00 9/13/00 copyright Brian Wil iams, 2000 4 Multiple Faults Occur courtesy of NASA three shorts, tank-line and pressure jacket burst, panel flies off. Divide & Conquer Diagnose each symptom. Summarize (conflicts) Combine APOLLO 13 Courtesy of Kanna Rajan, NASA Ames. Used with permission. 9/13/00 copyright Brian Wil iams, 2000 5 Due to the unknown mode, there tends to be an exponential number of diagnoses. U Candidates with UNKNOWN failure modes Good G Candidates with KNOWN failure modes Good F1 Fn G U Fault Models don’t help. 9/13/00 copyright Brian Wil iams, 2000 6 Diagnoses: (42 of 64 candidates) Fully Explained Failures ± [G(A),G(B),S0(C)] ± [G(A),S1(B),S0(C)] ± [S0(A),G(B),G(C)] . . . Fault Isolated, But Unexplained ± [G(A),G(B),U(C)] ± [G(A),U(B),G(C)] ± [U(A),G(B),G(C)] Partial Explained ± [G(A),U(B),S0(C)] ± [U(A),S1(B),G(C)] ± [S0(A),U(B),G(C)] . . . X Y A B C 0 0 0 0 in out 2
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3/6/00 9/13/00 copyright Brian Wil iams, 2000 7 Due to the unknown mode, there tends to be an exponential number of diagnoses. U Candidates with UNKNOWN failure modes Good G Candidates with KNOWN failure modes Good F1 Fn G U Fault Models don’t help. Most of the density space may be approximated by enumerating the few most likely diagnoses But these diagnoses represent a small fraction of the probability density space. Localizing a Robot within A Topological Map x 1 x 2 : p(x 2 |x 1 ,a)= .9 x 3 : p(x 3 |x 1 ,a)=.05 x 4 : p(x 4 |x 1 ,a)=.05 Observations can be features such as corridor features, junction features, etc. Posterior belief after an action An action is taken Posterior belief after sensing State Space Initial belief Estimating Dynamically with a Bayes Filter 3
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3/6/00 9/13/00 copyright Brian Wil iams, 2000 10 Outline Motivation Set Theoretic View of Propositional Logic From Propositional Logic to Probabilities Probabilistic Inference Methods Appenix: Application to Model-based Diagnosis 9/13/00 copyright Brian Wil iams, 2000 11 Propositional Logic Set Theoretic Semantics Universe of all interpretations (U) M(S) Set of all models for sentence S 9/13/00 copyright Brian Wil iams, 2000 12 Set Theoretic Semantics: True M(True) Universe U U 4
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3/6/00 9/13/00 copyright Brian Wil iams, 2000 13 Set Theoretic Semantics: False M(False) U ” f 9/13/00 copyright Brian Wil iams, 2000 14 Set Theoretic Semantics: not Q M(Q) M(True) U M(not Q) M(not Q) – M(Q) 9/13/00 copyright Brian Wil iams, 2000 15 Set Theoretic Semantics: Q and R M(R) M(Q) M(Q and R) M(Q and R) M(Q) ˙ M(R) U 5
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3/6/00 9/13/00 copyright Brian Wil iams, 2000 16
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l19bwintroprob - 3/6/00 Introduction to Probabilistic...

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