In Bayesian statistics, a credible interval (or Bayesian confidence interval) is an interval in the domain of a posterior probability distribution used for interval estimation. The generalisation to multivariate problems is the credible region. Credible intervals are analogous to confidence intervals in frequentist statistics, although they differ on a philosophical basis; 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.
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:
- Choosing the narrowest interval, which for a unimodal distribution will involve choosing those values of highest probability density including the mode.
- Choosing the interval where the probability of being below the interval is as likely as being above it. This interval will include the median.
- Assuming that the mean exists, choosing the interval for which the mean is the central point.
Contrasts with confidence interval
A frequentist 95% confidence interval of 35–45 means that with a large number of repeated samples, 95% of the calculated confidence intervals would include the true value of the parameter. The probability that the parameter is inside the given interval (say, 35–45) 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".
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; and also if the unknown parameter is a scale parameter (i.e. the forward probability function has the form ), with a Jeffreys' prior  — 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.
- Edwards, Ward, Lindman, Harold, Savage, Leonard J. (1963) "Bayesian statistical inference in psychological research". Psychological Review, 70, 193-242
- Lee, P.M. (1997) Bayesian Statistics: An Introduction, Arnold. ISBN 0-340-67785-6
- "Frequentism and Bayesianism".
- O'Hagan, A. (1994) Kendall's Advance Theory of Statistics, Vol 2B, Bayesian Inference, Section 2.51. Arnold, ISBN 0-340-52922-9
- Antelman, G. (1997) Elementary Bayesian Statistics (Madansky, A. & McCulloch, R. eds.). Cheltenham, UK: Edward Elgar ISBN 978-1-85898-504-6
- 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