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# lecture26 - Lecture 26 The Law of Likelihood Patrick Maher...

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Lecture 26 The Law of Likelihood Patrick Maher Philosophy 102 Spring 2009

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Favoring Notation: E = evidence; H i = incompatible hypotheses. Definition E favors H 1 over H 2 if p ( H 1 | E ) p ( H 2 | E ) > p ( H 1 ) p ( H 2 ) . In words: Learning E increases the ratio of the probability of H 1 to the probability of H 2 . Note E favors H 1 over H 2 ” doesn’t imply p ( H 1 | E ) > 1 / 2 or p ( H 1 | E ) > p ( H 1 ).
Examples 1 p ( H 1 | E ) = 0 . 2, p ( H 2 | E ) = 0 . 1, p ( H 1 ) = p ( H 2 ) = 0 . 4. p ( H 1 | E ) p ( H 2 | E ) = 2 > 1 = p ( H 1 ) p ( H 2 ) E favors H 1 over H 2 . 2 p ( H 1 | E ) = 0 . 6, p ( H 2 | E ) = 0 . 3, p ( H 1 ) = 0 . 4, p ( H 2 ) = 0 . 2. p ( H 1 | E ) p ( H 2 | E ) = 2 = p ( H 1 ) p ( H 2 ) E doesn’t favor H 1 over H 2 or H 2 over H 1 .

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The law of likelihood The law (proof at the end) E favors H 1 over H 2 if and only if p ( E | H 1 ) > p ( E | H 2 ). Probabilities of the form p ( E | H i ) are called likelihoods . Examples 1 A medical test has a false positive rate of 2% and a false negative rate of 3%. Let T = Sarah tested positive, D = Sarah has the disease. p ( T | D ) = 0 . 97 > 0 . 02 = p ( T |∼ D ) . By the law of likelihood, T favors D over D . 2 Sam is an Illinois crow. Let S = Sam is black, A = all crows are black, I = all Illinois crows are black.
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lecture26 - Lecture 26 The Law of Likelihood Patrick Maher...

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