Credible interval


In Bayesian statistics, a credible interval is an interval within which an unobserved parameter value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution. 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. Also, Bayesian credible intervals use knowledge of the situation-specific prior distribution, while the frequentist confidence intervals do not.
For example, in an experiment that determines the distribution of possible values of the parameter, if the subjective 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.

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. In frequentist terms, the parameter is fixed and the confidence interval is random.
Bayesian credible intervals can be quite different from 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, with a prior that is a uniform flat distribution; and also if the unknown parameter is a scale parameter, 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 cases; in general no such equivalence can be made.