Importance Sampling Example

#graph
curve(exp(-x+cos(x))*exp(-x), 0, 2.5, n = 101, add = FALSE, type = "l",
      ylab = NULL, log = NULL, lwd=3)
curve(exp(-x),0,2.5, add = TRUE, col = "violet", lwd=3)
curve(2*exp(-2*x),0,2.5, add = TRUE, col = "orange", lwd=3)

#importance sampling
m = 10000 #iterations

X = rexp(m, rate = 2)
weights = c()
mu = c()

for(i in 1:m){
  weights[i] = dexp(X[i])/dexp(X[i], rate = 2)
  mu[i] = weights[i]*exp((-1*(X[i]))+cos(X[i]))
}

#estimated mu
mean(mu)

## [1] 1.157902

sqrt(var(mu)/m)

## [1] 0.00268609

#naive MC
X  =  rexp(m, 1)

#Naive Monte Carlo mean
mean(exp(-1*X+cos(X)))

## [1] 1.176339

sqrt(var(exp(-1*X+cos(X)))/m)

## [1] 0.008860858