混合正态分布的重要性抽样

Question

How do you compute the mean of a gaussian mixture model via importance sampling? Say i have a model such that there is a 60% chance of being sampled from a N(-1,1) distribution, and a 40% chance of being sampled from a N(2,1/9) distribution. Below is what i have from the standard importance sampling format of g*f(x)/h; but i do not think my function f is correct because I used a summation mix function instead of truly sampling through probability. Is there any suggestion on what I should change? thanks!

set.seed(100)
N = 100
x = rnorm(N,mean=0,sd=0.6) # Sample x samples from the proposal distribution h(x)
h = dnorm(x,mean=0,sd=0.6) # Evaluate h(x)
g = rep(0,N)
g=x


f = function(x) {        # Evaluate f(x)
  f = 0.6*rnorm(x,-1,1)+0.4*rnorm(x,2,1/9)
  return(f)
} 

mean(g*f(x)/h)

Answer 1

For importance sampling, I guess there are several points that you should change in your code:

1. You may need a sufficient large N
2. You should use dnorm , rather than rnorm in your function f

An example is as below:

N = 1e8
X = rnorm(N) # Sample x samples from the proposal distribution h(x)
h = dnorm(X) # Evaluate h(x)


f = function(x) 0.6*dnorm(x,-1,1)+0.4*dnorm(x,2,1/9)      # Evaluate f(x)

mean(X*f(X)/h)
# [1] 0.2002296

As you can see, the theoretical mean of the mixed normal distribution is 0.6*(-1) + 0.4*2 = 0.2 , which is consistent to the result via importance sampling.