mixture importance sampling

Question

I have to integrate a 2 dimensional function between a range which has two peaks. I am trying to combine two Gaussian functions to get a distribution which is close to the function so as to use importance sampling to make a better estimate with lower variance. Do i have to draw from both the samples equally? My second question is how do i combine the samples drawn from both these distributions?

Answer 1

If, when targeting the density $g(x)$ you use an importance density function of the form $$\alpha f_1(x)+(1-\alpha) f_2(x)$$ which is called a mixture distribution you can either

1. simulate directly from the mixture, which amounts to simulate $$N_1\sim\mathcal{B}(n,\alpha)\quad x_1,\ldots,x_{N_1}\sim f_1\quad y_1,\ldots,y_{n-N_1}\sim f_2$$and use the importance weight$$\dfrac{g(x)}{\alpha f_1(x)+(1-\alpha) f_2(x)}$$for both the $x_i$'s and the $y_i$'s
2. set $1\le n_1\le n-1$ in an arbitrary manner and simulate $$x_1,\ldots,x_{n_1}\sim f_1\quad y_1,\ldots,y_{n-n_1}\sim f_2$$and use the importance weight$$\dfrac{n g(x)}{n_1 f_1(x)+(n-n_1) f_2(x)}$$for both the $x_i$'s and the $y_i$'s

since both approaches are correct. The second approach is described in Owen and Zhou (2000, JASA) and extended in our adaptive multiple mixture paper.