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###Credible intervals and confidence intervals are constructed in different ways and can be different

###Conclusion

###About the exception

###Additional:

Credible intervals and confidence intervals are constructed in different ways and can be different

Conclusion

About the exception

Additional:

The confidence interval is restricted in the way that it draws the boundaries. The confidence interval places these boundaries by considering the conditional distribution $X_\theta$ and will cover $\alpha \%$ independent from what the true value of $\theta$ is.

The confidence interval is restricted in the way that it draws the boundaries. The confidence interval places these boundaries by considering the conditional distribution $X_\theta$ and will cover $\alpha \%$ independent from what the true value of $\theta$ is (this independence is both the strength and weakness of the confidence interval).

The $\alpha \%$ confidence interval will correctly estimate/contain the true parameter $\alpha \%$ of the time, for a each parameter $\theta$. But for a given observation $X$ the $\alpha \%$ confidence interval will not estimate/contain the true parameter $\alpha \%$ of the time. (type I errors will occur at the same rate $\alpha \%$ for different values of the underlying parameter $\theta$. But for different observations $X$ the type I error rate will be different. For some observations the confidence interval may be more/less often wrong than for other observations).

The $\alpha \%$ credible interval will correctly estimate/contain the true parameter $\alpha \%$ of the time, for each observation $X$. But for a given parameter $\theta$ the $\alpha \%$ credible interval will not estimate/contain the true parameter $\alpha \%$ of the time. (type I errors will occur at the same rate $\alpha \%$ for different values of the observed parameter $X$. But for different underlying parameters $\theta$ the type I error rate will be different. For some underlying parameters the credible interval may be more/less often wrong than for other underlying parameters).

See in the image below the expression of conditional probability/chance of containing the parameter for this particular example

The $\alpha \%$ confidence interval will correctly estimate/contain the true parameter $\alpha \%$ of the time, for a each parameter $\theta$. But for a given observation $X$ the $\alpha \%$ confidence interval will not estimate/contain the true parameter $\alpha \%$ of the time. (type I errors will occur at the same rate $\alpha \%$ for different values of the underlying parameter $\theta$. But for different observations $X$ the type I error rate will be different. For some observations the confidence interval may be more/less often wrong than for other observations).

The $\alpha \%$ credible interval will correctly estimate/contain the true parameter $\alpha \%$ of the time, for each observation $X$. But for a given parameter $\theta$ the $\alpha \%$ credible interval will not estimate/contain the true parameter $\alpha \%$ of the time. (type I errors will occur at the same rate $\alpha \%$ for different values of the observed parameter $X$. But for different underlying parameters $\theta$ the type I error rate will be different. For some underlying parameters the credible interval may be more/less often wrong than for other underlying parameters).

Code for computing both images:

# parameters
set.seed(1)
n <- 2*10^4
perc = 0.95
za <- qnorm(0.5+perc/2,0,1)

# model
tau <- 1
theta <- rnorm(n,0,tau)
X <- rnorm(n,theta,1)

# plot scatterdiagram of distribution
plot(theta,X, xlab=expression(theta), ylab = "observed X",
     pch=21,col=rgb(0,0,0,0.05),bg=rgb(0,0,0,0.05),cex=0.25,
     xlim = c(-5,5),ylim=c(-5,5)
    )

# confidence interval
t <- seq(-6,6,0.01)
lines(t,t-za*1,col=2)
lines(t,t+za*1,col=2)

# credible interval
obsX <- seq(-6,6,0.01)
lines(obsX*tau^2/(tau^2+1)+za*sqrt(tau^2/(tau^2+1)),obsX,col=3)
lines(obsX*tau^2/(tau^2+1)-za*sqrt(tau^2/(tau^2+1)),obsX,col=3)

# adding contours for joint density
conX <- seq(-5,5,0.1)
conT <- seq(-5,5,0.1)
ln <- length(conX)

z <- matrix(rep(0,ln^2),ln)
for (i in 1:ln) {
  for (j in 1:ln) {
    z[i,j] <- dnorm(conT[i],0,tau)*dnorm(conX[j],conT[i],1)
  }
}
contour(conT,conX,-log(z), add=TRUE, levels = 1:10 )

legend(-5,5,c("confidence interval","credible interval","log joint density"), lty=1, col=c(2,3,1), lwd=c(1,1,0.5),cex=0.7)
title(expression(atop("scatterplot and contourplot of", 
                      paste("X ~ N(",theta,",1)   and   ",theta," ~ N(0,",tau^2,")"))))




# expression succes rate as function of X and theta
# Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?
layout(matrix(c(1:2),1))
par(mar=c(4,4,2,2),mgp=c(2.5,1,0))
pX <- seq(-5,5,0.1)
pt <- seq(-5,5,0.1)
cc <- tau^2/(tau^2+1)

plot(-10,-10, xlim=c(-5,5),ylim = c(0,1),
     xlab = expression(theta), ylab = "chance of containing the parameter")
lines(pt,pnorm(pt/cc+za/sqrt(cc),pt,1)-pnorm(pt/cc-za/sqrt(cc),pt,1),col=3)
lines(pt,pnorm(pt+za,pt,1)-pnorm(pt-za,pt,1),col=2)
title(expression(paste("for different values ", theta)))

legend(-3.8,0.15,
       c("confidence interval","credible interval"),
       lty=1, col=c(2,3),cex=0.7, box.col="white")


plot(-10,-10, xlim=c(-5,5),ylim = c(0,1),
     xlab = expression(X), ylab = "chance of containing the parameter")
lines(pX,pnorm(pX*cc+za*sqrt(cc),pX*cc,sqrt(cc))-pnorm(pX*cc-za*sqrt(cc),pX*cc,sqrt(cc)),col=3)
lines(pX,pnorm(pX+za,pX*cc,sqrt(cc))-pnorm(pX-za,pX*cc,sqrt(cc)),col=2)
title(expression(paste("for different values ", X)))


text(0,0.3, 
     c("95% Confidence Interval\ndoes not imply\n95% chance of containing the parameter"),
     cex= 0.7,pos=1)

library(shape)
Arrows(-3,0.3,-3.9,0.38,arr.length=0.2)