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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.

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    That could be a measure of the change in uncertainty given the observation, though there are others. It might not be particularly meaningful if you had started with an improper prior – 2017-02-22

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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.]