Credible Interval

A credible interval is the Bayesian counterpart to the frequentist confidence interval. It provides a range within which an unknown parameter falls with a specified posterior probability, given the observed data and a prior distribution. A 95% credible interval means that, conditional on the data and the prior, there is a 95% probability that the true parameter value lies within that range. This interpretation is more intuitive than that of a confidence interval, which refers to the long run frequency of intervals containing the true value across hypothetical repetitions of the study.

As Bayesian methods become more common in clinical research, epidemiology, and the social sciences, credible intervals appear with increasing frequency in published studies. Understanding what they mean and how they differ from confidence intervals is essential for interpreting Bayesian analyses. Credible intervals answer a question many researchers and clinicians want to ask: "Given the data, what is the probable range of the parameter?" This directness makes Bayesian results particularly appealing for clinical decision making, where stakeholders want to know the probability that a treatment effect falls within a clinically meaningful range.

Bayesian analysis begins with a prior distribution that represents existing knowledge or beliefs about the parameter before observing the data. The likelihood function summarizes the information in the observed data. Bayes' theorem combines the prior and the likelihood to produce the posterior distribution, which represents updated knowledge about the parameter. The credible interval is derived from this posterior distribution. The highest posterior density (HPD) interval is the narrowest interval containing the specified probability mass. An equal tailed interval places equal probability in each tail; for a 95% interval, its limits are the 2.5th and 97.5th percentiles of the posterior, the default summary in many Bayesian software packages. For a symmetric, unimodal posterior the two intervals coincide; they differ when the posterior is skewed. When the prior is noninformative (flat or vague), the credible interval often numerically resembles the frequentist confidence interval, but the interpretation remains categorically different.

Report credible intervals whenever you conduct a Bayesian analysis, whether in a Bayesian clinical trial, a Bayesian meta analysis, or a Bayesian hierarchical model. Credible intervals are particularly useful when incorporating prior information is appropriate (such as historical control data in a clinical trial), when the posterior distribution is asymmetric or multimodal (making symmetric confidence intervals misleading), or when the research question explicitly concerns the probability that a parameter lies in a specific range. Many adaptive and platform trial designs use Bayesian decision criteria, usually thresholds on posterior probabilities (such as the probability that a treatment is superior), reported alongside credible intervals.

An important mistake is interpreting a credible interval as if it were a confidence interval, or vice versa. A 95% credible interval states that the probability of the parameter falling within the interval is 0.95, conditional on the data and prior; a 95% confidence interval states that if the experiment were repeated infinitely, 95% of such intervals would contain the true value. Another error is choosing an unduly informative prior that dominates the data, producing a credible interval that reflects prior beliefs more than observed evidence. Researchers should always report the prior used and conduct sensitivity analyses to show how results change with alternative priors. Presenting a Bayesian credible interval without specifying whether it is an HPD interval or an equal tailed interval creates ambiguity.

A clinical trial uses a Bayesian adaptive design to evaluate a new antifungal agent. The prior distribution for the treatment effect is based on two earlier phase II trials. After enrolling 300 patients, the posterior distribution for the difference in cure rates (new drug minus standard of care) has a mean of 12.4 percentage points and a 95% equal tailed credible interval of 4.8 to 19.6 percentage points. The posterior probability that the new drug is superior is greater than 99.9%. Because the entire credible interval exceeds the prespecified clinical significance threshold of 3 percentage points, the data monitoring committee recommends stopping the trial for efficacy.