10 Mar 2004
CS 3243 - Uncertainty
12
Conditional probability
nDefinition of conditional probability:
nP(a | b) = P(a Ù b) / P(b) if  P(b) > 0
n
nProduct rule gives an alternative formulation:
nP(a Ù b) = P(a | b) P(b) = P(b | a) P(a)
n
nA general version holds for whole distributions, e.g.,
nP(Weather,Cavity) = P(Weather | Cavity) P(Cavity)
n(View as a set of 4 × 2 equations, not matrix mult.)
n
nChain rule is derived by successive application of product rule:
nP(X1, …,Xn) = P(X1,...,Xn-1) P(Xn | X1,...,Xn-1)
n                 = P(X1,...,Xn-2) P(Xn-1 | X1,...,Xn-2) P(Xn | X1,...,Xn-1)
n                  = …
n                  = πi= 1 P(Xi | X1, … ,Xi-1)
n