Vaniver comments on Conservation of Expected Evidence

Vaniver 7 May 2012 23:56 UTC
7 points
P(H) = P(H|E)P(E) + P(H|~E)P(~E)

P(H)*(P(E)+P(~E))=P(H|E)P(E) + P(H|~E)P(~E)

P(H)P(E)+P(H)P(~E)=P(H|E)P(E) + P(H|~E)P(~E)

P(H)P(~E)=(P(H|E)-P(H))*P(E) + P(H|~E)P(~E)

0=(P(H|E)-P(H))*P(E) + (P(H|~E)-P(H))*P(~E)

The trick is that P(E)+P(~E)=1, and so you can multiply the left side by the sum and the right side by 1.