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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,BC) = P(AC)P(BC) Are the presence of mushrooms
P(mushroom, anchovy  pepperoni): and anchovies x 1/5 NO!
1/5 = 3/5 independent given pepperoni
P(mushroom, =epperoniAP ) = 1/5 P (A, 1/3P ) /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 (AP ) 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 (AB ) 34 Bayes Theorem P (A, B ) = P (A) ∗ P (B A)
= P (B ) ∗ P (AB ) Bayes Theorem
P (AB ) =
(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 = imageiA = tiger)P (A = tiger)
P (A = tigerB = 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 nonterrorists carry
swiss army knifes. What’s the probability that a knife carrier is a terrorist?
P (terr) = .000001 prior probability
P (knifeterr) = .5 likelihood
P (knife ∼ terr) = .1
compute
P (knife) = P (terr) ∗ P (knifeterr) + P (∼ terr) ∗ P (knife ∼ terr)
= .1
P (knifeterr) ∗ P (terr)
P (terrknife) =
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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