Comparison of the efficiency of hit-miss method with that of simple mean-value method 1 Hit-miss method rarest)hit-miss c(b-a)]drg(xHI 2) Simple mean-value method: Ⅴa(gs)=N(ba)gg()r Since c gx), var(I est/hit-miss var est simple The simple mean-value method is more efficient than the hit-miss method Remark Different Monte Carlo integration methods do not have the same efficiency
Comparison of the efficiency of hit-miss method with that of simple mean-value method 1) Hit-miss method = − − I − I b a hit miss c b a dxg x est N 2 ( ) ( ) 1 var( ) 2) Simple mean-value method: = − − I I b a simple b a dxg x g x est N 2 ( ) ( ) ( ) 1 var( ) Since c g(x), var(Iest)hit-miss> var(Iest)simple. The simple mean-value method is more efficient than the hit-miss method. Remark: Different Monte Carlo integration methods do not have the same efficiency
How to improve the efficient of Monte carlo integration method? One widely used method is based on the variance reduction Importance sampling method. Basic ideal 与agx)h8x1x)( f(x)>0 Intuitively, one expects a small variance when g(x)/f(x)is nearly constant Variance var 8(x)d fx g( f When g(x>0 and fxh Ax, var[g(x)/f(x)]=0 In more general cases, when Nx-gtxy, var[g(x)f(x)] reaches a minimum
How to improve the efficient of Monte Carlo integration method? One widely used method is based on the variance reduction. Importance sampling method: Basic ideal: f f x g x f x f x g x I dxg x dx ( ) ( ) ( ) ( ) ( ) = ( )= = f(x) > 0 Intuitively, one expects a small variance when g(x)/f(x) is nearly constant. Variance: I g f x x dx f x g x 2 2 ( ) ( ) ( ) ( ) var = − When g(x)>0 and = ( ) ( ) ( ) dxg x g x f x , var[g(x)/f(x)]=0. In more general cases, when = ( ) ( ) ( ) dxg x g x f x , var[g(x)/f(x)] reaches a minimum