### R commands for Section 6.4.1 -- Importance Sampling
### G. Givens & J. Hoeting, Computational Statistics, 2nd edition, Wiley, 2013.


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# IMPORTANCE SAMPLING
#
# Key idea:  Let X~exponential(1).  Use importance sampling to find
# E(sqrt(x)).   Compare the performance of IS using various envelopes.
# Some code is commented out for use in exercises in 6.1 below.

# Importance sampling (weights are not standardized)
samp.size=10000

## Envelopes
##Case 1: abs(Normal(0,1))
set.seed(1)
x=abs(rnorm(samp.size))
wts=dexp(x)/(dnorm(x)/.5)

# Importance sampling estimate of the mean
mu.is = (1/samp.size)*sum(sqrt(x)*wts)
mu.is  # 0.8535775

#variance of the weights and effective sample size
var(wts)  # 0.5812052
samp.size/(1+var(wts)) #ESS  # 6324.29

#Importance sampling with standardized weights, estimated mean
mu.sis = sum(sqrt(x)*wts)/sum(wts)
mu.sis    # 0.8654587


##Case 2: uniform(0,1000)
set.seed(1)
x=runif(samp.size,0,1000)
wts=dexp(x)/(1/1000)

# Importance sampling estimate of the mean
mu.is = (1/samp.size)*sum(sqrt(x)*wts)
mu.is   # 1.03526

#variance of the weights and effective sample size
var(wts) # 492.2098
samp.size/(1+var(wts)) #ESS  # 20.27535

#Importance sampling with standardized weights, estimated mean
mu.sis = sum(sqrt(x)*wts)/sum(wts)
mu.sis   # 0.9489134


##Case 3: abs(Cauchy(0,1))
set.seed(1)
x=abs(rcauchy(samp.size))
wts=dexp(x)/(dcauchy(x)/.5)

# Importance sampling estimate of the mean
mu.is = (1/samp.size)*sum(sqrt(x)*wts)
mu.is   # 0.8803712

#variance of the weights and effective sample size
var(wts)   # 0.1787769
samp.size/(1+var(wts)) #ESS  # 8483.37

#Importance sampling with standardized weights, estimated mean
mu.sis = sum(sqrt(x)*wts)/sum(wts)
mu.sis     # 0.8776175