Conditional Probability
P(A|B) = Fraction of worlds in which B is true that also have A true.
H = “Have a headache”
F = “Coming down with Flu”
P(H) = 1/10
P(F) = 1/40
P(H|F) = 1/2
“Headaches are rare and flu is rarer, but if you’re coming down with ‘flu there’s a 50-50 chance you’ll have a headache.”
P(H|F) = Fraction of flu-inflicted worlds in which you have a headache.
Definition of Conditional Probability.
P(A|B) = P(A ^ B) / P(B)
Corollary: The Chain Rule.
P(A ^ B) = P(A|B) P(B)
The Bayes Rule:
P(B|A) = P(A ^ B) / P(A) = P(A|B) P(B) / P(A)
Suppose A can take on more than 2 values.
A is a random variable with arity k if it can take on exactly one value out of {v1,v2, .. vk}.
Thus...
P(A=vi ∧ A=vj)=0 if i≠ j
P(A=v1∨ A=v2 ∨ A=vk) =1
From the previous axioms:
P(B∧[A=v1 ∨A=v2 ∨A=vi]) = ∑ P(B∧A=vj )
And thus we can prove
P(B) = ∑ P(B ∧ A = vj )
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