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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 to fall within it. For example, in an experiment that determines the distribution of possible values of the parameter , if the probability that lies between 35 and 45 is 0.95, then is a 95% credible interval.
Credible intervals are typically used to characterize posterior probability distributions or predictive probability distributions. [1] The generalisation to multivariate problems is the 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.
Credible intervals are not unique; any given posterior probability distribution has an infinite number of 95% credible interval. There are therefore multiple methods for defining a suitable credible interval:
For multi-dimensional problems, the highest posterior density region is bounded by a probability density contour line. [4]
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 ), with a Jeffreys' prior [6] — 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 (albeit important) cases; in general no such equivalence can be made.