n <- 220

s <- 100

print( theta.hat.mle <- s / n )

# [1] 0.4545455

# the Beta( alpha, beta ) density (for alpha > 0, beta > 0) is 

#   c * theta^( alpha - 1 ) * ( 1 - theta )^( beta - 1 )

# the likelihood function is 

#   c * theta^s * ( 1 - theta )^( n - s ),

# which is the Beta( s + 1, n - s + 1 ) distribution;
# here that's the Beta( 101, 121 ) distribution

# the posterior density is Beta( alpha + s, beta + n - s )

# the neutral prior is Beta( 1, 1 ) = Uniform on the interval ( 0, 1 )

# the cut-the-DA-slack prior is Beta( 17.4, 4.6 )

# with the neutral prior, the posterior is Beta( 101, 121 )

# with the cut-the-DA-slack prior, the posterior is Beta( 117.4, 124.6 )

# make the prior-likelihood-posterior plot with the neutral prior

theta.grid.1 <- seq( 0, 1, length = 500 )

plot( theta.grid.1, dbeta( theta.grid.1, 101, 121 ), type = 'l',
  xlab = 'theta', ylab = 'Density', lwd = 2, 
  main = 'Neutral Prior', ylim = c( 0, 12.5 ) )

lines( theta.grid.1, dbeta( theta.grid.1, 1, 1 ), lty = 3, lwd = 2,
  col = 'blue' )

lines( theta.grid.1, dbeta( theta.grid.1, 101, 121 ), lty = 2, lwd = 2,
  col = 'red' )

text( 0.1, 2, 'prior', col = 'blue' )

text( 0.3, 11, 'likelihood' )

text( 0.6, 10, 'posterior', col = 'red' )

segments( 0.791, 0, 0.791, 1.5, lwd = 2 )

text( 0.791, 2.25, 'no discrimination' )

# make the prior-likelihood-posterior plot with the
# cut-the-DA-slack prior

plot( theta.grid.1, dbeta( theta.grid.1, 101, 121 ), type = 'l',
  xlab = 'theta', ylab = 'Density', lwd = 2, 
  main = 'Cut-the-DA-Slack Prior', ylim = c( 0, 12.5 ) )

lines( theta.grid.1, dbeta( theta.grid.1, 17.4, 4.6 ), lty = 3, lwd = 2,
  col = 'blue' )

lines( theta.grid.1, dbeta( theta.grid.1, 117.4, 124.6 ), 
  lty = 2, lwd = 2, col = 'red' )

text( 0.8, 5.5, 'prior', col = 'blue' )

text( 0.3, 11, 'likelihood' )

text( 0.6, 10, 'posterior', col = 'red' )

segments( 0.791, 0, 0.791, 1.5, lwd = 2 )

text( 0.791, 2.25, 'no discrimination' )

# superimpose the two posterior distributions on the same graph
# to see how sensitive the posterior is to the prior

plot( theta.grid.1, dbeta( theta.grid.1, 101, 121 ), type = 'l',
  xlab = 'theta', ylab = 'Density', lwd = 2, ylim = c( 0, 12.5 ),
  col = 'red', main = 'Sensitivity Analysis' )

lines( theta.grid.1, dbeta( theta.grid.1, 117.4, 124.6 ), 
  lty = 2, lwd = 2, col = 'red' )

text( 0.3, 11, 'neutral', col = 'red' )

text( 0.7, 10, 'cut-the-DA-slack', col = 'red' )

segments( 0.791, 0, 0.791, 1.5, lwd = 2 )

text( 0.791, 2.25, 'no discrimination' )