Importance Sampling This is still unbiased Y]-∫Y(x)px)dk Ω ,1111111, E(f(x)) p(x)dx =∫f(x)d Ω X1 XN for all N
Importance Sampling • This is still unbiased x1 xN E(f(x)) = = = f x dx p x dx p x f x E Y Y x p x dx i ( ) ( ) ( ) ( ) ( ) ( ) for all N
Importance Sampling Zero variance if p(x)f(x) p(x)=cf(x) E(f(x)) p(x,)c Var(Y)=0 Less variance with better X1 XN importance sampling
Importance Sampling • Zero variance if p(x) ~ f(x) x1 xN E(f(x)) Less variance with better importance sampling ( ) 0 1 ( ) ( ) ( ) ( ) = = = = Var Y p x c f x Y p x cf x i i i
Generating Random Points Uniform distribution: Use pseudorandom number generator 1 2
Generating Random Points • Uniform distribution: – Use pseudorandom number generator Probability 0 1