Quick example demonstrating the importance sampling idea.

importance-sampling-example.py

# 
import scipy as sc
from scipy.stats import norm

N=100
M = 1000
nsamp = sc.zeros(M)

for i in range(M):
        data=sc.randn(N)*0.1
        nsamp[i] = sc.mean(data<-0.5)

print "Direct estimation:",sc.mean(nsamp)

# Importance sampling
p = lambda x: norm.pdf(x,scale=0.1) # original distribution
q = lambda x: norm.pdf(x,loc=-0.5) # proposal distribution --- focuses on the regime of interest!

# new quantity of interest
f = lambda x: (x<-0.5)*p(x)/q(x)

for i in range(M):
        data = -0.5+sc.randn(N)
        nsamp[i] = sc.mean(f(data))

print "Importance sampling:", sc.mean(nsamp)
print "Exact value:", norm.cdf(-0.5,scale=0.1)