# Credible interval

A credible interval is an interval within which an unobserved parameter value falls with a particular posterior probability, computed from the posterior distribution of a Bayesian analysis.<sup>[1](https://user.math.uzh.ch/vorlesungen/sta120/fs25/web/1.%20Lecture/12%20Bayesian%20Methods.pdf)</sup> It is the Bayesian analog of the frequentist confidence interval, but its interpretation differs: a 95% credible interval is an interval in the domain of a posterior probability distribution (or predictive distribution) such that, given the data and the prior, the probability that the parameter lies in the interval is 0.95.<sup>[2](https://www.youtube.com/watch?v=kMRG7Q0p66w)</sup><sup> • </sup><sup>[3](https://www.fda.gov/regulatory-information/search-fda-guidance-documents/guidance-use-bayesian-statistics-medical-device-clinical-trials)</sup> The generalization to multivariate parameter spaces is called a credible region.<sup>[2](https://www.youtube.com/watch?v=kMRG7Q0p66w)</sup>

| Key fact | Detail |
|---|---|
| Definition | An interval \( (a, b) \) such that \( P(a \le \theta \le b \mid x) = 1 - \alpha \), where the probability is taken over the posterior distribution of \( \theta \)<sup>[4](https://www2.stat.duke.edu/~rcs46/books/bayes_manuscripts.pdf)</sup> |
| Interpretation | Probability is over the parameter, given the data; a confidence interval's probability is over the data, for all parameter values<sup>[5](https://www.stat.ucla.edu/~ywu/Bayesian.pdf)</sup> |
| Standard constructions | Equal-tailed (percentile) interval and highest posterior density (HPD) interval; they coincide only when the posterior is symmetric<sup>[6](https://www.casrai.org/guides/summarizing-a-posterior-distribution)</sup> |
| Regulatory use | FDA strongly encourages reporting of credible intervals for Bayesian trials in the labeling of medical devices<sup>[3](https://www.fda.gov/regulatory-information/search-fda-guidance-documents/guidance-use-bayesian-statistics-medical-device-clinical-trials)</sup> |
| Coverage caveat | Bayesian credible intervals do not come with frequentist coverage guarantees; pronounced discrepancies between credibility and coverage can occur<sup>[7](https://www.sciencedirect.com/science/article/abs/pii/S0167715214003691)</sup> |
| Diffuse-prior limit | With a uniform prior, the equal-tailed Bayesian credible interval does not generally coincide with the frequentist confidence interval; any agreement depends on the model and the interval construction<sup>[8](https://www.cambridge.org/core/journals/spanish-journal-of-psychology/article/bayesian-estimation-with-informative-priors-is-indistinguishable-from-data-falsification/FFAB96BDC5EE3C64B144ECF8F90F31E9)</sup><sup> • </sup><sup>[16](https://mathworld.wolfram.com/CredibleInterval.html)</sup> |

## Definition and interpretation

Formally, a Bayesian credible interval of size \( 1 - \alpha \) is an interval \( (a, b) \) such that

\[ P(a \le \theta \le b \mid x) = \int_a^b p(\theta \mid x) \, d\theta = 1 - \alpha. \]

The definition does not determine a unique interval: many pairs \( (a, b) \) can enclose the same posterior mass, which is why specific constructions such as the equal-tailed and highest-density rules exist.<sup>[4](https://www2.stat.duke.edu/~rcs46/books/bayes_manuscripts.pdf)</sup> In discrete settings exact coverage may be impossible, so the definition is often stated with \( \ge \) rather than \( = \).<sup>[4](https://www2.stat.duke.edu/~rcs46/books/bayes_manuscripts.pdf)</sup>

The interpretation is the main point of contrast with the confidence interval. In the frequentist framework, the 95% figure describes the procedure: if the experiment were repeated many times, about 95% of the computed intervals would contain the true value, and any single realized interval either contains it or does not.<sup>[5](https://www.stat.ucla.edu/~ywu/Bayesian.pdf)</sup><sup> • </sup><sup>[4](https://www2.stat.duke.edu/~rcs46/books/bayes_manuscripts.pdf)</sup> In the Bayesian framework, probability is a state of knowledge conditional on the observed data, so the statement "there is a 95% probability that \( \theta \) lies in the interval, given the data" is available directly.<sup>[5](https://www.stat.ucla.edu/~ywu/Bayesian.pdf)</sup> A thought experiment makes the difference concrete: if a reliable source states that the true speed of light lies outside a computed interval, the Bayesian posterior probability that the interval contains the true value drops immediately, while the frequentist coverage probability, a property of the procedure, does not update.<sup>[5](https://www.stat.ucla.edu/~ywu/Bayesian.pdf)</sup>

## Equal-tailed and highest-density intervals

**Equal-tailed (percentile) intervals.** The equal-tailed interval, also called the central posterior interval, places \( \alpha/2 \) probability in each tail: for a 95% interval, the endpoints are the 2.5th and 97.5th percentiles of the posterior.<sup>[6](https://www.casrai.org/guides/summarizing-a-posterior-distribution)</sup><sup> • </sup><sup>[3](https://www.fda.gov/regulatory-information/search-fda-guidance-documents/guidance-use-bayesian-statistics-medical-device-clinical-trials)</sup> It is simple to compute from posterior draws, requiring only two quantiles, and it is invariant under any monotonic reparameterization: if the parameter is logged, exponentiated, or reciprocated, the interval on the new scale is just the old endpoints transformed the same way. It is always a single, easy-to-report range.<sup>[6](https://www.casrai.org/guides/summarizing-a-posterior-distribution)</sup> Its weakness appears with skewed posteriors: because it insists on equal tail mass regardless of density, some values inside the interval can have lower posterior density than some values outside it.<sup>[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC6630113/)</sup>

**Highest density intervals.** The highest posterior density (HPD) interval, or highest density interval (HDI), is the narrowest interval containing the specified posterior mass. For a unimodal posterior, every point inside the interval has higher posterior density than every point outside it, and the density at the two endpoints is equal.<sup>[6](https://www.casrai.org/guides/summarizing-a-posterior-distribution)</sup> This has two results: at any fixed probability level, the equal-tailed interval for a given posterior is at least as wide as its HDI, and when the posterior is truly multimodal, the HDI may split into several distinct intervals instead of a single unbroken span—something no equal-tailed interval can express.<sup>[6](https://www.casrai.org/guides/summarizing-a-posterior-distribution)</sup> The trade-off is that the HDI is not invariant under nonlinear reparameterization: computing the HDI and then transforming it does not generally match computing the HDI directly on the transformed scale.<sup>[6](https://www.casrai.org/guides/summarizing-a-posterior-distribution)</sup> Unimodality of the posterior is a sufficient condition for the HPD credible set to be a single interval, though not a necessary one.<sup>[4](https://www2.stat.duke.edu/~rcs46/books/bayes_manuscripts.pdf)</sup>

For a symmetric unimodal posterior the two constructions are nearly identical and the choice is effectively stylistic. For a skewed posterior they genuinely disagree: the HDI shifts toward the mode and is narrower, while the equal-tailed interval extends further into the long tail. Reading an HDI from one paper against an equal-tailed interval from another, without either stating which was used, compares two different quantities that share the label "95% credible interval."<sup>[6](https://www.casrai.org/guides/summarizing-a-posterior-distribution)</sup>

## Computation from posterior samples

Modern Bayesian computation usually relies on [Markov chain Monte Carlo](https://www.edgepedia.org/markov-chain-monte-carlo) (MCMC), which produces correlated draws from the posterior; interval endpoints are then estimated as quantiles of those draws.<sup>[10](https://mc-stan.org/docs/2_28/stan-users-guide/error-statistics-from-the-bootstrap.html)</sup> Because the draws are autocorrelated, uncertainty in estimated quantities is governed by the effective sample size rather than the raw number of draws, and the [Monte Carlo](https://www.edgepedia.org/monte-carlo) standard error of an estimate scales as \( 1/\sqrt{N_{\mathrm{eff}}} \).<sup>[11](https://mc-stan.org/docs/reference-manual/analysis.html)</sup> Tail quantiles are the hardest summaries to estimate: accurate 97.5% quantile estimates require roughly twenty times as many Monte Carlo simulations as the median.<sup>[10](https://mc-stan.org/docs/2_28/stan-users-guide/error-statistics-from-the-bootstrap.html)</sup> Research on interval estimation from MCMC output includes Rao-Blackwellized estimators that combine conditional-based calculations across iterations, and Chen and Shao's algorithm, which finds all possible 100(1−α)% credible intervals from an MCMC sample and proves convergence results for unimodal posteriors.<sup>[12](https://www.sciencedirect.com/science/article/abs/pii/S0378375802003270)</sup>

## Relation to frequentist coverage

A credible interval's posterior probability and its frequentist coverage, the proportion of repeated trials in which the interval contains the true value, are distinct properties and need not agree.<sup>[13](https://academic.oup.com/mnras/article/510/3/3207/6449384)</sup> For estimating a bounded normal mean with known variance, situations of pronounced discrepancy between the credibility of Bayes credible regions and their frequentist coverage have been exhibited; conversely, some frequentist confidence intervals have credibility one in certain cases.<sup>[7](https://www.sciencedirect.com/science/article/abs/pii/S0167715214003691)</sup> The two frameworks connect in special cases. Using a uniform prior does not guarantee that the frequentist confidence interval becomes the equal-tailed Bayesian credible interval; agreement depends on the model and interval construction.<sup>[8](https://www.cambridge.org/core/journals/spanish-journal-of-psychology/article/bayesian-estimation-with-informative-priors-is-indistinguishable-from-data-falsification/FFAB96BDC5EE3C64B144ECF8F90F31E9)</sup><sup> • </sup><sup>[16](https://mathworld.wolfram.com/CredibleInterval.html)</sup> In astronomy, matching priors have been constructed under which credible intervals can be interpreted approximately as confidence intervals, containing the truth a certain proportion of the time in repeated trials, a useful property when the data covariance matrix is itself an estimate.<sup>[13](https://academic.oup.com/mnras/article/510/3/3207/6449384)</sup>

## Use in regulated research and reporting practice

Regulators treat credible intervals as a standard summary of Bayesian analyses. The FDA's guidance for Bayesian medical device trials defines the 95% credible interval as an interval with posterior probability 0.95, recommends summarizing the posterior with a few numbers such as the posterior mean, standard deviation, and credible interval, and strongly encourages reporting credible intervals in device labeling.<sup>[3](https://www.fda.gov/regulatory-information/search-fda-guidance-documents/guidance-use-bayesian-statistics-medical-device-clinical-trials)</sup> A 2026 draft guidance extends this framework to drugs and biologics, where Bayesian calculations can govern interim analyses, inform dose selection, or support primary inference.<sup>[14](https://www.fda.gov/regulatory-information/search-fda-guidance-documents/use-bayesian-methodology-clinical-trials-drug-and-biological-products)</sup>

Reporting guidelines address the ambiguity among interval constructions. The Bayesian Analysis Reporting Guidelines ask authors to state explicitly whether they report density-based values (mode and HDI) or quantile-based values (median and equal-tailed interval), and to state the interval's mass, for example 95%; in most cases the marginal posterior is unimodal and only modestly skewed, so the two summaries are quantitatively similar except for highly skewed or multimodal distributions with deep troughs.<sup>[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC8526359/)</sup> Practical guidance adds that the interval type should be named explicitly, since an unqualified "95% CI" for a Bayesian result reads as a frequentist confidence interval by convention, and that neither construction on a skewed posterior is generally symmetric around the mean, median, or mode.<sup>[6](https://www.casrai.org/guides/summarizing-a-posterior-distribution)</sup>

## References

1. [Bayesian Methods, University of Zurich lecture notes](https://user.math.uzh.ch/vorlesungen/sta120/fs25/web/1.%20Lecture/12%20Bayesian%20Methods.pdf)
2. [Credible interval, WikiAudio (CC-BY-SA text)](https://www.youtube.com/watch?v=kMRG7Q0p66w)
3. [Guidance for the Use of Bayesian Statistics in Medical Device Clinical Trials, U.S. FDA](https://www.fda.gov/regulatory-information/search-fda-guidance-documents/guidance-use-bayesian-statistics-medical-device-clinical-trials)
4. [A First Course in Bayesian Statistical Methods (manuscript), Duke University](https://www2.stat.duke.edu/~rcs46/books/bayes_manuscripts.pdf)
5. [A Modern Introduction to Bayesian Statistics, UCLA](https://www.stat.ucla.edu/~ywu/Bayesian.pdf)
6. [Summarizing a Posterior Distribution, CASRAI](https://www.casrai.org/guides/summarizing-a-posterior-distribution)
7. [On the discrepancy between Bayes credibility and frequentist probability of coverage, Statistics & Probability Letters (2015)](https://www.sciencedirect.com/science/article/abs/pii/S0167715214003691)
8. [Bayesian Estimation with Informative Priors is Indistinguishable from Data Falsification, Spanish Journal of Psychology](https://www.cambridge.org/core/journals/spanish-journal-of-psychology/article/bayesian-estimation-with-informative-priors-is-indistinguishable-from-data-falsification/FFAB96BDC5EE3C64B144ECF8F90F31E9)
9. [Understanding and interpreting confidence and credible intervals around effect estimates, Brazilian Journal of Physical Therapy](https://pmc.ncbi.nlm.nih.gov/articles/PMC6630113/)
10. [Error statistics from the bootstrap, Stan User's Guide](https://mc-stan.org/docs/2_28/stan-users-guide/error-statistics-from-the-bootstrap.html)
11. [Posterior Analysis, Stan Reference Manual](https://mc-stan.org/docs/reference-manual/analysis.html)
12. [Estimating Bayesian credible intervals, Journal of Statistical Planning and Inference (2003)](https://www.sciencedirect.com/science/article/abs/pii/S0378375802003270)
13. [Matching Bayesian and frequentist coverage probabilities when using an approximate data covariance matrix, MNRAS (2022)](https://academic.oup.com/mnras/article/510/3/3207/6449384)
14. [Use of Bayesian Methodology in Clinical Trials of Drug and Biological Products (draft guidance), U.S. FDA](https://www.fda.gov/regulatory-information/search-fda-guidance-documents/use-bayesian-methodology-clinical-trials-drug-and-biological-products)
15. [Bayesian Analysis Reporting Guidelines](https://pmc.ncbi.nlm.nih.gov/articles/PMC8526359/)
16. [Credible Interval -- from Wolfram MathWorld](https://mathworld.wolfram.com/CredibleInterval.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Bayesian statistics › Bayesian probability and inference foundations › Prior and posterior analysis*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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