Credible Interval

A credible interval is a range that, in Bayesian statistics, contains an unknown quantity with a stated probability. Say your model estimates a conversion rate and reports a ninety-five percent credible interval of two to five percent. The Bayesian reading is refreshingly direct, because given the data you have and the assumptions you started with, there is a ninety-five percent probability the true rate lies between two and five percent. The interval comes from the posterior distribution, the updated picture of what the parameter could be after the data has been folded in with your prior belief. You take the posterior and carve out a central band that captures the stated share of the probability. That band is the credible interval, and it says exactly what most people intuitively want an interval to say.

Because it rests on a posterior, a credible interval depends on two ingredients, the data and the prior. The prior encodes what you believed about the parameter before seeing this data — perhaps from earlier experiments, perhaps a deliberately vague stance that lets the data speak. Combine prior and data through Bayes' rule and you get the posterior, and the credible interval is simply a summary of it. This is why two analysts can compute different credible intervals from the same data, because they brought different priors. With plenty of data the prior's influence fades and the interval is driven mostly by the evidence; with little data the prior matters a great deal. A credible interval is therefore an honest statement of belief conditioned on assumptions, not a claim that pretends to have none.

The credible interval is constantly confused with its frequentist cousin, the confidence interval, and the difference is more than jargon. A ninety-five percent credible interval means there is a ninety-five percent probability the parameter lies in this particular range, given the data. A ninety-five percent confidence interval means something subtler and often misunderstood, because if you repeated the whole experiment many times and built an interval each time by the same procedure, about ninety-five percent of those intervals would contain the true value. The confidence statement is about the long-run behavior of the method, not about this one interval. So the everyday interpretation people reach for — a ninety-five percent chance the truth is in here — is actually the credible interval's meaning, not the confidence interval's, even though the confidence interval is the one most often reported.

In practice the two can land in similar places. With a lot of data and a vague prior, a credible interval and a confidence interval for the same parameter often nearly coincide, which is why the distinction is easy to gloss over. But they answer different questions and can diverge sharply when the prior is informative or the data is thin. A credible interval treats the parameter as uncertain and the data as fixed, and reports probability about the parameter. A confidence interval treats the parameter as a fixed unknown and the data as the random thing, and reports probability about the procedure. Which to use depends on your framework and whether you have a prior worth using. The mistake to avoid is stating a confidence interval and then interpreting it as if it were a credible one.

To use credible intervals well, be explicit about the prior, since it shapes the result. Choose it deliberately — a weak, uninformative prior when you want the data to dominate, or a stronger one when you have real prior evidence and want to use it — and be ready to show that the conclusion does not hinge on an arbitrary choice. Report the interval alongside the posterior it came from, and say what probability level it covers. Read it for what it is, a direct probability statement about the parameter given your data and assumptions, which makes it easy to communicate to non-statisticians who naturally think in those terms. And when the data is sparse, lean on the prior consciously rather than pretending the interval is assumption-free, because with little evidence the prior is doing much of the work.

The failures come from muddling the two frameworks or hiding the prior. Interpreting a confidence interval as though it were a credible interval is the most common error, common enough that even experienced analysts slip. Choosing a prior that quietly forces the answer, then presenting the credible interval as objective, is a subtler abuse. Reporting an interval without its probability level, or without noting how much the prior influenced it, leaves the reader unable to judge it. And treating a wide credible interval as a failure misses the point, because a wide interval honestly reflects genuine uncertainty, and narrowing it dishonestly with overconfident priors only trades honesty for false precision. The discipline is to state the prior, state the probability level, and let the credible interval mean exactly what it says.

Credible comes from Latin credibilis (believable, from credere, to believe), and interval from intervallum (space between), naming a believable range for a parameter.

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