Credible interval

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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 ${\displaystyle \alpha }$ to fall within it. For example, in an experiment that determines the distribution of possible values of the parameter ${\displaystyle \mu }$, if the probability that ${\displaystyle \mu }$ lies between 35 and 45 is ${\displaystyle \alpha =0.95}$, then ${\displaystyle 35\leq \mu \leq 45}$ is a 95% credible interval.

Credible intervals are typically used to characterize posterior probability distributions or predictive probability distributions.^[1] Their generalization to disconnected or multivariate sets is called 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.