In Bayesian statistics, a credible interval is a posterior probability interval^{[1]} which is used for interval estimation in contrast to point estimation. Credible intervals are used for purposes similar to those of confidence intervals in frequentist statistics and an alternative terminology is to use Bayesian confidence interval instead of "credible interval".^{[2]} When dealing with more than one unknown quantity simultaneously, the term credible region is used.
For example, a statement such as "following the experiment, a 90% credible interval for the parameter t is 3545" means that the posterior probability that t lies in the interval from 35 to 45 is 0.9.
There are several ways of defining a credible interval from a given probability distribution for the parameter. Examples 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.^{[3]}
A frequentist 90% confidence interval of 3545 means that with a large number of repeated samples, 90% of the 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 nonrandom 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 confidence interval as "... one interval generated by a procedure that will give correct intervals 95 % [resp. 90 %] of the time". ^{[4]}
In general, Bayesian credible intervals do not coincide with frequentist confidence intervals for two reasons:
Many professional statisticians and decisions scientists as well as nonstatisticians intuitively interpret confidence intervals in the Bayesian credible interval sense and hence "credible intervals" are sometimes called "confidence intervals". It is widely accepted, especially in the decision sciences, that "credible interval" is merely the subjective subset of "confidence intervals". In fact, much research in calibrated probability assessments never uses the term "credible interval" and it is common to simply use "confidence interval".

