lecture24.pdf - COMPSCI 240 Reasoning Under Uncertainty...

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COMPSCI 240: Reasoning Under Uncertainty Arya Mazumdar University of Massachusetts at Amherst Fall 2016
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Lecture 24: Review
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Model of Probability A Sample Space Ω Probability Law: A Ω; P ( A ) Events Sets Modeling: More likely event to get more probability
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Axioms of Probability Nonnegativity: P ( A ) 0 Additivity: For any two disjoint sets A , B , P ( A B ) = P ( A ) + P ( B ) Holds for infinitely many disjoints events A 1 , A 2 , A 3 , . . . P ( i A i ) = X i P ( A i ) . Normalization: P (Ω) = 1
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Basic identities P ( ) = 0 P ( A c ) = 1 - P ( A ) If A B , then P ( A ) P ( B ). P ( A B ) = P ( A ) + P ( B ) - P ( A B ) Sub-additivity: P ( A B ) P ( A ) + P ( B )
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New probability space P ( ·| B ) Define, P ( A | B ) = P ( A B ) P ( B ) . In the case of disjoint A and B , A B = . Which means, P ( A B ) = 0. So P ( A | B ) = 0. Supports our intuition.
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New probability space P ( ·| B ) Conditional probability: a valid probability measure Axioms of probability are satisfied. New sample space is B . Nonnegativity: P ( A | B ) 0 Normalization: P ( B | B ) = P ( B B ) P ( B ) = P ( B ) P ( B ) = 1.
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Multiplication Rule Shorthand: P ( n i =1 A i ) P ( A 1 A 2 . . . A n ) The Rule: P ( n i =1 A i ) = P ( A 1 ) P ( A 2 | A 1 ) P ( A 3 | A 1 A 2 ) . . . P ( A n | ∩ n - 1 i =1 A i )
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Law of total probability Let A 1 , A 2 , . . . , A n partition Ω and P ( A i ) > 0 P ( B ) = P ( A 1 B ) + P ( A 2 B ) + · · · + P ( A n B ) = P ( A 1 ) P ( B | A 1 ) + P ( A 2 ) P ( B | A 2 ) + · · · + P ( A n ) P ( B | A n ) . Recall the disjoint additivity rule.
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Bayes’ Rule Let A 1 , A 2 , . . . , A n partition Ω and P ( A i ) > 0. For any B such that P ( B ) > 0, P ( A i | B ) = P ( A i ) P ( B | A i ) P ( B ) = P ( A i ) P ( B | A i ) P ( A 1 ) P ( B | A 1 ) + P ( A 2 ) P ( B | A 2 ) + · · · + P ( A n ) P ( B | A n )
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Discrete Probability Laws If Ω is finite and all outcomes are equally likely, then P ( A ) = | A | | Ω | .
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Discrete Probability Laws If Ω is finite and all outcomes are equally likely, then P ( A ) = | A | | Ω | . Sometimes it’s challenging to compute | A | and | Ω | and they are too large work out by hand. . .
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Shortcuts for Counting Permutations: There are n ! = n × ( n - 1) × . . . × 2 × 1 ways to permute n objects. E.g., permutations of { a , b , c } are abc , acb , bac , bca , cab , cba
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Shortcuts for Counting Permutations: There are n ! = n × ( n - 1) × . . . × 2 × 1 ways to permute n objects. E.g., permutations of { a , b , c } are abc , acb , bac , bca , cab , cba k -Permutations: There are n × ( n - 1) × . . . × ( n - k + 1) ways to choose the first k elements of a permutation of n objects. E.g., 2-permutations of { a , b , c , d } are ab , ac , ad , ba , bc , bd , ca , cb , cd , da , db , dc
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Shortcuts for Counting Permutations: There are n ! = n × ( n - 1) × . . . × 2 × 1 ways to permute n objects. E.g., permutations of { a , b , c } are abc , acb , bac , bca , cab , cba k -Permutations: There are n × ( n - 1) × . . . × ( n - k + 1) ways to choose the first k elements of a permutation of n objects. E.g., 2-permutations of { a , b , c , d } are ab , ac , ad , ba , bc , bd , ca , cb , cd , da , db , dc Combinations: There are ( n k ) = n ! k !( n - k )!
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