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 and confidence regions 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 (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 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. 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 (the maximum a posteriori). This is sometimes called the highest posterior density interval (HPDI). Choosing the interval where the probability of being below the interval is as likely as being above it. This interval will include the median. This is sometimes called the equal-tailed interval. Assuming that the mean exists, choosing the interval for which the mean is the central point. It is possible to frame the choice of a credible interval within decision theory and, in that context, a smallest interval will always be a highest probability density set. It is bounded by the contour of the density. Credible intervals can also be estimated through the use of simulation techniques such as Markov chain Monte Carlo.