##############################################################################################################
#
# Figure1.4 (page 32) – Inadequacy of the SIR method when the overlap between the proposal density 
#                       and the target density is relatively small.
#
# p(x) = 0.7*N(x;5,6.25) + 0.3*N(x;0,4)
# q(x) = t(-5,1.44,2)
#
##############################################################################################################
#
# Author : Hedibert Freitas Lopes
#          Graduate School of Business
#          University of Chicago
#          5807 South Woodlawn Avenue
#          Chicago, Illinois, 60637
#          Email : hlopes@ChicagoGSB.edu
#
###############################################################################################################
set.seed(13883)
M = 500
par(mfrow=c(1,1))
x  = seq(-15,15,length=5000)
p  = 0.7*dnorm(x,5,2.5)+0.3*dnorm(x,0,2)
q = 0.694*dt((x+5)/1.2,2)
plot(x,p,xlab="",ylab="",main="",ylim=c(0,0.25),type="l",axes=F)
axis(1)
axis(2,at=round(seq(0,0.25,length=6),2))
lines(x,q,lty=1)
text(-3.5,0.225,"q",cex=2)
text(7.5,0.1,expression(pi),cex=2)

# Sampling from q
# ---------------
x1 = -5+1.2*rt(M,2)
for (i in 1:M)
  segments(x1[i],0,x1[i],-0.005)

# Weighted resampling
# -------------------
w = (0.7*dnorm(x1,5,2.5)+0.3*dnorm(x1,0,2))/(0.694*dt((x1+5)/1.2,2))
w = w/sum(w)
for (i in 1:M)
  segments(x1[i],0,x1[i],0.75*w[i],lty=1)
abline(h=0)