lecture25 - Lecture 25 Bayess Theorem Patrick Maher...

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Lecture 25 Bayes’s Theorem Patrick Maher Philosophy 102 Spring 2009
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Bayes’s theorem Suppose H is a hypothesis, E is some evidence, and we want to calculate p ( H | E ). Often p ( E | H ) is easier to calculate than p ( H | E ). Bayes’s theorem allows us to calculate p ( H | E ) from p ( E | H ) and other probabilities. It is named after Thomas Bayes (c. 1702–1761). The theorem: simple case p ( H | E ) = p ( E | H ) p ( H ) p ( E | H ) p ( H ) + p ( E |∼ H ) p ( H ) .
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Dice example An urn contains 10 dice, of which 9 are fair but 1 is biased so it lands six half the time. You randomly draw a die from the urn and toss it. Let S = the die lands six, B = the die is biased. p ( B | S ) = p ( S | B ) p ( B ) p ( S | B ) p ( B ) + p ( S |∼ B ) p ( B ) = (1 / 2)(1 / 10) (1 / 2)(1 / 10) + (1 / 6)(9 / 10) = 1 / 4 .
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Mammography example A woman has a lump in her breast; her physician examines it and believes the probability is 0.01 that it is cancer. However, the physician orders a mammogram and the result is positive. The probability of a positive result from a mammogram is 0.8 given
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lecture25 - Lecture 25 Bayess Theorem Patrick Maher...

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