Credible interval
A range that genuinely has, say, a 95% probability of holding the true value.
What it means
A credible interval is a range that carries a stated posterior probability — say 95% — of holding the true value, given the prior and model. It is the Bayesian counterpart to the confidence interval, but built from the posterior distribution rather than the sampling behaviour of a procedure. Because it is derived from the posterior distribution, it permits — provided the prior and model are correct — the intuitive statement people wrongly attach to confidence intervals: that there is, for instance, a 95% probability the parameter lies within it. Common constructions take either the central 95% of the posterior (equal-tailed) or the narrowest span holding that mass (highest-density). Since the interval inherits the prior, it is only as defensible as that prior, and a misspecified model can yield a confidently wrong range.
How it is built
The recipe is mechanical once you have a posterior distribution, the probability distribution over the parameter after prior and likelihood are combined. You pick a probability mass, say 95%, and carve out a region of the parameter axis holding that much of the posterior's area. Two conventions dominate. An equal-tailed interval takes the 2.5th and 97.5th percentiles, leaving equal probability in each tail; it always brackets the median and is simple to report. A highest-density interval instead takes the narrowest span containing the mass, so every point inside is more probable than any point outside. On a symmetric posterior the two agree; on a skewed one they diverge, and the equal-tailed version can exclude the most probable values while keeping less probable ones in its long tail.
The interpretation frequentists cannot make
The whole appeal is a licence the confidence interval withholds. A credible interval is a statement about the parameter given the data: after seeing these results, 95% of the posterior probability sits in this range. A confidence interval is a statement about the procedure: across hypothetical repetitions, 95% of intervals built this way would trap the fixed true value, which says nothing direct about the single interval in front of you. The distinction is not pedantic hair-splitting. When Hoekstra and colleagues surveyed students and working researchers, both groups endorsed false statements that read a confidence interval as though it were a credible interval, and more experience barely helped. The credible interval simply says the thing people already believe they are saying.
What the prior contributes
Because the interval is carved from the posterior, it inherits whatever the prior brought. With a flat or weakly informative prior and a reasonable amount of data, the posterior is dominated by the likelihood, and the credible interval lands almost exactly where the matching confidence interval does, a convergence formalized by the Bernstein-von Mises theorem. With little data, or a deliberately informative prior, the two part company: a sceptical prior pulls the interval toward the null and narrows it, an enthusiastic one does the opposite. This is a feature rather than a defect, since the interval reports assumptions and evidence together, but it means a credible interval is only as defensible as the prior behind it, and stating that prior is part of reporting the interval.
Where it breaks down
The intuitive 95% is conditional: it holds only if the model and prior are right, so a badly misspecified model yields a confidently wrong interval. The highest-density interval carries its own quirks. It is not preserved under nonlinear transformation, so the interval for a variance is not simply the squared interval for a standard deviation, whereas equal-tailed intervals transform cleanly. When the posterior is multimodal the highest-density region can split into disconnected pieces that no single interval represents honestly. And because most posteriors are approximated by simulation, the endpoints carry Monte Carlo noise that shrinks with more samples but never fully vanishes. None of this undermines the concept; it just means the clean probability statement rests on assumptions worth stating out loud.
Examples
A 95% credible interval of [1, 7] for an effect means the posterior assigns 95% probability to the true effect lying between 1 and 7.
An election forecaster reports a 95% credible interval of 44 to 52 seats. Unlike a confidence interval, this licenses the plain statement people want: 95% probability the true count lies inside.
Two analysts read the same trial data, one with a sceptical prior and one with an optimistic one, and report different credible intervals — the range encodes assumptions as well as evidence.
A Bayesian vaccine trial reports a 95% credible interval for efficacy of 90% to 98%. The team can say directly that the posterior puts 95% probability on efficacy falling in that band.
An A/B test's posterior gives a 95% credible interval of -0.3% to +1.8% for the conversion lift, so a product team can say the change is probably positive but might do nothing.
First described in Bayesian tradition; Lindley and others.
Key references
- Makowski, D., Ben-Shachar, M. S., & Lüdecke, D. (2019). bayestestR: Describing Effects and their Uncertainty, Existence and Significance within the Bayesian Framework. Journal of Open Source Software, 4(40), 1541. doi.org/10.21105/joss.01541
- Hespanhol, L., Vallio, C. S., Costa, L. M., & Saragiotto, B. T. (2019). Understanding and interpreting confidence and credible intervals around effect estimates. Brazilian Journal of Physical Therapy, 23(4), 290-301. doi.org/10.1016/j.bjpt.2018.12.006
- Kruschke, J. K., & Liddell, T. M. (2018). The Bayesian New Statistics: Hypothesis testing, estimation, meta-analysis, and power analysis from a Bayesian perspective. Psychonomic Bulletin & Review, 25(1), 178-206. doi.org/10.3758/s13423-016-1221-4
- Morey, R. D., Hoekstra, R., Rouder, J. N., Lee, M. D., & Wagenmakers, E.-J. (2016). The fallacy of placing confidence in confidence intervals. Psychonomic Bulletin & Review, 23(1), 103-123. doi.org/10.3758/s13423-015-0947-8
- Hoekstra, R., Morey, R. D., Rouder, J. N., & Wagenmakers, E.-J. (2014). Robust misinterpretation of confidence intervals. Psychonomic Bulletin & Review, 21(5), 1157-1164. doi.org/10.3758/s13423-013-0572-3