Credible interval
It is defined such that an unobserved parameter value has a particular probability
For example, in an experiment that determines the distribution of possible values of the parameter
[1] Their generalization to disconnected or multivariate sets is called credible region.
[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.
Credible regions are not unique, as any given probability distribution has an infinite number of
For example, in the univariate case, there are multiple definitions for a suitable interval or region: One may also define an interval for which the mean is the central point, assuming that the mean exists.
-SCR) can easily be generalized to the multivariate case, and are bounded by probability density contour lines.
Credible intervals can also be estimated through the use of simulation techniques such as Markov chain Monte Carlo.
[5] 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 (cannot be considered to have a distribution of possible values) and the confidence interval is random (as it depends on the random sample).
Bayesian credible intervals differ from frequentist confidence intervals by two major aspects: 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 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;[6] and also if the unknown parameter is a scale parameter (i.e. the forward probability function has the form
But these are distinctly special (albeit important) cases; in general no such equivalence can be made.
