Lecture 4-Probability Slides

Ucsdedutriesch 12 32 conditional independence venn

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Unformatted text preview: sd.edu/~triesch 12 32 Conditional Independence Venn diagrams in Pizza form from B. Warner Conditional Independence: P(A,B|C) = P(A|C)P(B|C) Are the presence of mushrooms P(mushroom, anchovy | pepperoni): and anchovies x 1/5 NO! 1/5 = 3/5 independent given pepperoni P(mushroom, =epperoniA|P ) = 1/5 P (A, 1/3|P ) /3 x /5 NO! P (M |P ) p 3/5 P ( | anchovy): M = 2 = 1 1/3 P(anchovy, pepperoni | mushroom): 1/4 = 2/4 x 3/4 NO! so P (A, M |P ) is not equal to P (M |P ) × P (A|P ) therefor the presence of mushrooms with pizza that h not conditionally independent Quiz: come upand anchovies areas conditional independence (given the presence of pepperoni). Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 12 Bayes Theorem P (A, B ) = P (A) ∗ P (B |A) 33 Bayes Theorem P (A, B ) = P (A) ∗ P (B |A) = P (B ) ∗ P (A|B ) 34 Bayes Theorem P (A, B ) = P (A) ∗ P (B |A) = P (B ) ∗ P (A|B ) Bayes Theorem P (A|B ) = (proof above) P (B |A)P (A) P (B ) 35 36 Bayes Theorem Terminology Bayes’ Theorem: posterior probability likelihood prior probability P( B | A) P( A) P( A | B) ! P( B) evidence P(B) is often computed as Note 1: likelihood and prior probability are often much easier to measure than posterior probability, so it makes sense to express the latter as a function of (B ) = P the former. B |Ai)P (Ai) P( i Note 2: P(B) typically also expressed as function of conditionals and priors: where Ai are all possible disjoint subsets P ( B | Ai ) P( Ai ) P( Ai | B) ! " P( B | Ai ) P( Ai ) i Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 15 Bayes Rule in Biology Given an image of an animal you have to determine whether the animal is a tiger or not P (B = imagei|A = tiger)P (A = tiger) P (A = tiger|B = imagei) = P (B = imagei) 37 Sample Problem (numbers made up) The probability that an individual at the airport is a terrorist is 1 in 10 million (1 × 10−6) Half the terrorists carry swiss army knifes. 10% of non-terrorists carry swiss army knifes. What’s the probability that a knife carrier is a terrorist? P (terr) = .000001 prior probability P (knife|terr) = .5 likelihood P (knife| ∼ terr) = .1 compute P (knife) = P (terr) ∗ P (knife|terr) + P (∼ terr) ∗ P (knife| ∼ terr) = .1 P (knife|terr) ∗ P (terr) P (terr|knife) = P (knife) = .5 ∗ .000001/.1 = 5 ∗ 10−6 38 Continuous probability density For continuous random variables it does not make sense to talk about the probability of a particular value (which is equal to 0) Instead we talk about probability density p(x) is a probability density over a continuous variable b P r(x ∈ [a, b]) = p(x)dx a e.g. probability density of heights of females 39 Bayes Rule revisited We can still have Bayes rule for continuous...
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This note was uploaded on 02/09/2014 for the course COGS 109 taught by Professor Staff during the Fall '08 term at UCSD.

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