Credible Interval on the Prior Does it make sense to describe a credible interval on the prior? I know that it's defined in terms of the posterior, but is there an equivalent notion on the prior. Is there a separate term for it?
E.g., I want to compare the width of the 95% credible interval on the prior vs. the width on the posterior. Does that make sense at all to do?
I'm sorry if this is vague, I'm not sure how to phrase it better. 
 A: Without getting into quibbles about the terminology 'credible interval', I believe it
is entirely reasonable to compare the prior and posterior distributions in
the way you suggest.
Suppose you are seeking a Bayesian interval estimate for parameter $\theta,$ considered (in the Bayesian manner) as a random variable. Maybe
$\theta$ is the success probability in a binomial process and you
are considering the distribution $\mathsf{Beta}(330,270)$ as the prior.
What might lead to this choice? Maybe, according to prior experience or belief,
you think that $\theta$ is "above 0.5, but not likely above 0.6." Then
this is a reasonable prior on several grounds: its mean, median, and mode
are all about $0.55.$ Also, this distribution puts about 95% of its
probability in the interval $(0.51, 0.59)$.
330/(270+330)
## 0.55                        # mean
qbeta(.5, 330, 270)
## 0.5500556                   # median
qbeta(c(.025,.975), 330, 270)
## 0.5100824 0.5896018

Later, after observing the binomial process through 1000 trials and counting
620 successes, we combine a likelihood for these data with the prior
distribution to obtain the posterior distribution $\mathsf{Beta}(960, 650).$
The posterior has mean about $0.594.$ Also, it puts about 95% of its probability
in $(0.57, 0.62),$  which is the 95% Bayesian posterior credible interval. 
The data have changed our view: the success probability seems higher now than
we supposed when we chose the prior. It is only natural, perhaps inevitable, to compare the 95% probability
interval $(0.51, 0.59)$ from the prior distribution with the 95% probability interval $(0.57, 0.62)$ from the posterior distribution. [I will leave the 
discussion whether it is proper to call
both intervals 'credible' intervals up to the definitions of various
textbooks on Bayesian inference.]
