The highest-density 90% credible interval of a probability distribution

In Bayesian statistics, a credible interval is an interval used to characterize a probability distribution. It is defined such that an unobserved parameter value has a particular probability to fall within it. For example, in an experiment that determines the distribution of possible values of the parameter , if the probability that lies between 35 and 45 is 0.95, then is a 95% credible interval.

Credible intervals are typically used to characterize posterior probability distributions or predictive probability distributions.[1] The generalisation to multivariate problems is the credible region.

Credible intervals are a Bayesian analog to confidence intervals in frequentist statistics.[2] The two concepts arise from different philosophies:[3] Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat their bounds as random variables and the parameter as a fixed value. Also, Bayesian credible intervals use (and indeed, require) knowledge of the situation-specific prior distribution, while the frequentist confidence intervals do not.

Choosing a credible interval

Credible intervals are not unique; any given posterior probability distribution has an infinite number of 95% credible intervals. There are therefore multiple methods for defining a suitable credible interval:

For multi-dimensional problems, the highest posterior density region is bounded by a probability density contour line.[4]

Credible intervals can also be estimated through the use of simulation techniques such as Markov chain Monte Carlo.[5]

Contrasts with confidence interval

See also: Confidence interval § Credible interval

A frequentist 95% confidence interval means that with a large number of repeated samples, 95% of such calculated confidence intervals would include the true value of the parameter. In frequentist terms, the parameter is fixed (cannot be considered to have a distribution of possible values) and the confidence interval is random (as it depends on the random sample).

Bayesian credible intervals differ from frequentist confidence intervals by two major aspects:

For the case of a single parameter and data that can be summarised in a single sufficient statistic, it can be shown that the credible interval and the confidence interval coincide if the unknown parameter is a location parameter (i.e. the forward probability function has the form ), with a prior that is a uniform flat distribution;[6] and also if the unknown parameter is a scale parameter (i.e. the forward probability function has the form ), with a Jeffreys' prior   [6] — the latter following because taking the logarithm of such a scale parameter turns it into a location parameter with a uniform distribution. But these are distinctly special (albeit important) cases; in general no such equivalence can be made.


  1. ^ Edwards, Ward; Lindman, Harold; Savage, Leonard J. (1963). "Bayesian statistical inference in psychological research". Psychological Review. 70 (3): 193–242. doi:10.1037/h0044139.
  2. ^ Lee, P.M. (1997) Bayesian Statistics: An Introduction, Arnold. ISBN 0-340-67785-6
  3. ^ VanderPlas, Jake. "Frequentism and Bayesianism III: Confidence, Credibility, and why Frequentism and Science do not Mix | Pythonic Perambulations".
  4. ^ O'Hagan, A. (1994) Kendall's Advanced Theory of Statistics, Vol 2B, Bayesian Inference, Section 2.51. Arnold, ISBN 0-340-52922-9
  5. ^ Chen, Ming-Hui; Shao, Qi-Man (1 March 1999). "Monte Carlo Estimation of Bayesian Credible and HPD Intervals". Journal of Computational and Graphical Statistics. 8 (1): 69–92. doi:10.1080/10618600.1999.10474802.
  6. ^ a b Jaynes, E. T. (1976). "Confidence Intervals vs Bayesian Intervals", in Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, (W. L. Harper and C. A. Hooker, eds.), Dordrecht: D. Reidel, pp. 175 et seq

Further reading