Variance Reduction Techniques Problem:variance decreases with 1/N Increasing samples removes noise slowly Variance reduction: -Stratified sampling -Importance sampling
Variance Reduction Techniques • Problem: variance decreases with 1/N – Increasing # samples removes noise slowly • Variance reduction: – Stratified sampling – Importance sampling
Stratified Sampling o Estimate subdomains separately Arvo Ek(f(x)) 0 0 0 0 0 X1 XN 0 0
Stratified Sampling • Estimate subdomains separately x1 xN Ek (f(x)) Arvo
Stratified Sampling This is still unbiased -2 艺以 E(f(x)) X1 XN
Stratified Sampling • This is still unbiased = = = = M k i i N i N i N F N f x N F 1 1 1 ( ) 1 x1 xN Ek (f(x))
Stratified Sampling Less overall variance if less variance in subdomains Varr] Ei(f(X) X1 XN
Stratified Sampling • Less overall variance if less variance in subdomains = = M k N Ni Var Fi N Var F 1 2 1 x1 xN Ek (f(x))
Importance Sampling Put more samples where f(x)is bigger j恤 E(f(x)) =) p(x,) X1 XN
Importance Sampling • Put more samples where f(x) is bigger ( ) ( ) 1 ( ) 1 i i i N i i p x f x Y Y N f x dx = = = x1 xN E(f(x))