##############################################################################################################
#
#  Example 2.1 (pages 44, 48 and 50) and Example 2.3 (pages 48-49) – Figure 2.1 (page 49) 
# 
#  Normal data with unknown mean and known variance.
#
#                 Example 2.1 – Normal prior for the unknown mean
#                 Example 2.3 – Cauchy prior for the unknown mean
#
##############################################################################################################
#
# Author : Hedibert Freitas Lopes
#          Graduate School of Business
#          University of Chicago
#          5807 South Woodlawn Avenue
#          Chicago, Illinois, 60637
#          Email : hlopes@ChicagoGSB.edu
#
###############################################################################################################

# Sufficient statistics
xbar  = 7.0
sig2n = 4.5

# Hyperparameters of the normal prior
mu0   = 0.0
tau20 = 1.0

# Hyperparameters of the conjugate normal posterior
tau21 = 1/(1/sig2n+1/tau20)
mu1   = tau21*(xbar/sig2n+mu0/tau20)

# Figure 2.1
# ----------
x = seq(-4,14,length=5000)
f.normal = dnorm(x,mu1,sqrt(tau21))
f.normal = f.normal/max(f.normal)*0.0045
f.cauchy = dnorm(x,xbar,sqrt(sig2n))/(tau20+(x-mu0)^2)
f.cauchy = f.cauchy/max(f.cauchy)*0.0015
par(mfrow=c(1,1))
plot(x,f.normal,xlab="",ylab="",main="",type="l",axes=F)
axis(1,at=c(-4,0,5,10,14))
axis(2)
lines(x,f.cauchy,lty=2)