Credible interval

In Bayesian statistics, a credible interval is an interval in the domain of a posterior probability distribution or predictive distribution used for interval estimation.[1] The generalisation to multivariate problems is the credible region. Credible intervals are analogous to confidence intervals in frequentist statistics,[2] although they differ on a philosophical basis;[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.

For example, in an experiment that determines the uncertainty distribution of parameter , if the probability that lies between 35 and 45 is 0.95, then is a 95% credible interval.

Choosing a credible interval

Credible intervals are not unique on a posterior distribution. Methods for defining a suitable credible interval include:

It is possible to frame the choice of a credible interval within decision theory and, in that context, an optimal interval will always be a highest probability density set.[4]

Contrasts with confidence 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. The probability that the parameter is inside the given interval (say, 3545) is either 0 or 1 (the non-random unknown parameter is either there or not). 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). Antelman (1997, p. 375) summarizes a [95%] confidence interval as "... one interval generated by a procedure that will give correct intervals 95% of the time".[5]

In general, Bayesian credible intervals do not coincide with frequentist confidence intervals for two reasons:

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 will 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, 193-242
  2. Lee, P.M. (1997) Bayesian Statistics: An Introduction, Arnold. ISBN 0-340-67785-6
  3. "Frequentism and Bayesianism".
  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. Antelman, G. (1997) Elementary Bayesian Statistics (Madansky, A. & McCulloch, R. eds.). Cheltenham, UK: Edward Elgar ISBN 978-1-85898-504-6
  6. 1 2 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
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