BernGridSerie4b = function( Theta , pTheta , Data ,mycex=1.4 ) {
# Bayesian updating for Bernoulli likelihood and prior specified on a grid.
# Input arguments:
#  Theta is a vector of theta values, all between 0 and 1.
#  pTheta is a vector of corresponding probability _masses_.
#  Data is a vector of 1's and 0's, where 1 corresponds to a and 0 to b.
# Output:
#  pThetaGivenData is a vector of posterior probability masses over Theta.
#  Also creates a three-panel graph of prior, likelihood, and posterior 
#  probability masses with credible interval.


# Create summary values of Data
z = sum( Data==1 ) # number of 1's in Data
N = length( Data ) # number of flips in Data
# Compute the likelihood of the Data for each value of Theta.
pDataGivenTheta = Theta^z * (1-Theta)^(N-z)
# Compute the evidence pData and the posterior.
pData = sum( pDataGivenTheta * pTheta )
pThetaGivenData = pDataGivenTheta * pTheta / pData

dotsize <- 8 # how big to make the plotted dots

#--------------- Plot the prior.
meanTheta = sum( Theta * pTheta ) # mean of prior, for plotting
plot( Theta , pTheta , type="p" , pch="." , cex=dotsize ,
      xlim=c(0,1) , ylim=c(0,1.1*max(pThetaGivenData)) , cex.axis=1.2 ,
      xlab=bquote(theta) , ylab=bquote(p(theta)) , cex.lab=1.5 ,
      main="Prior" , cex.main=1.5 )
if ( meanTheta > .5 ) {
   textx = 0 ; textadj = c(0,1)
} else {
  textx = 1 ; textadj = c(1,1)
}
text( textx , 1.0*max(pThetaGivenData) ,
      bquote( "mean(" * theta * ")=" * .(signif(meanTheta,3)) ) ,
      cex=mycex , adj=textadj )

#------------- Plot the likelihood: p(Data|Theta)
plot(Theta ,pDataGivenTheta ,type="p" ,pch="." ,cex=dotsize
	,xlim=c(0,1) ,cex.axis=1.2 ,xlab=bquote(theta) 
	,ylim=c(0,1.1*max(pDataGivenTheta)) 
	,ylab=bquote( "p(D|" * theta * ")" )  
	,cex.lab=1.5 ,main="Likelihood" ,cex.main=1.5 )
if ( z > .5*N ) { textx = 0 ; textadj = c(0,1) }
else { textx = 1 ; textadj = c(1,1) }
text( textx ,1.0*max(pDataGivenTheta) ,cex=mycex
	,bquote( "Data: z=" * .(z) * ",N=" * .(N) ) ,adj=textadj )

#------------ Plot the posterior.
meanThetaGivenData = sum( Theta * pThetaGivenData )
plot(Theta ,pThetaGivenData ,type="p" ,pch="." ,cex=dotsize
	,xlim=c(0,1) ,ylim=c(0,1.1*max(pThetaGivenData)) ,cex.axis=1.2 
	,xlab=bquote(theta) ,ylab=bquote( "p(" * theta * "|D)" )
	,cex.lab=1.5 ,main="Posterior" ,cex.main=1.5 )
if ( meanThetaGivenData > .5 ) { textx = 0 ; textadj = c(0,1) } 
else { textx = 1 ; textadj = c(1,1) }
text(textx ,1.00*max(pThetaGivenData) ,cex=mycex
	,bquote( "mean(" * theta * "|D)=" * .(signif(meanThetaGivenData,3)) ) 
	,adj=textadj )

#----- evidence
text(textx ,0.75*max(pThetaGivenData) ,cex=mycex
	,bquote( "p(D)=" * .(signif(pData,3)) ) ,adj=textadj )


return( pThetaGivenData )
} # end of function