Credible Interval

A range of values, calculated within a Bayesian framework, believed to contain a parameter's true value with a specified posterior probability.

Source: O'Hagan & Stevens 2001

A credible interval, calculated within a Bayesian framework, is a range that is said to contain a parameter's true value with a stated posterior probability, such as ninety-five per cent. It permits the natural interpretation that the parameter has that probability of lying within the range, given the data and the prior, which is exactly what people often wrongly assume a confidence interval means. It emerges from the posterior distribution, so the prior influences it alongside the data. Stating a probable range for the parameter itself is what it offers. Sutton and Abrams (2001) describe this.

Source: Sutton & Abrams 2001

A credible interval is interpreted directly as a probability statement about the parameter: a ninety-five per cent credible interval means that, given the data and the prior, there is a ninety-five per cent posterior probability that the parameter lies within the interval. This differs from a confidence interval, whose probability refers to the procedure over repeated samples. So a credible interval is interpreted as containing the parameter with the stated posterior probability, which is the intuitive interpretation many mistakenly apply to confidence intervals, and this directness is a feature of the Bayesian approach, in which parameters have probability distributions and statements about them are made directly.

Source: O'Hagan & Stevens 2001

A credible interval is calculated from the posterior distribution of the parameter, by finding a range that contains the specified posterior probability, commonly either the central interval between the relevant percentiles of the posterior or the highest posterior density interval, which is the shortest range containing that probability. So a credible interval is calculated by summarising the posterior distribution, taking a range that holds the desired probability mass, which requires the posterior, obtained by combining the prior with the data through Bayes' theorem, and is often computed from posterior samples generated by methods such as Markov chain Monte Carlo when the posterior has no simple closed form.

Source: O'Hagan & Stevens 2001

A credible interval, from Bayesian analysis, is interpreted as containing the parameter with a stated posterior probability given the data and prior, while a confidence interval, from frequentist analysis, is interpreted through its long-run coverage, the proportion of such intervals over repeated samples that would contain the fixed parameter. The credible interval makes a direct probability statement about the parameter; the confidence interval does not. So the two differ in interpretation and framework, with the credible interval treating the parameter probabilistically and the confidence interval treating it as fixed, and although they can coincide numerically in some cases, their meanings are distinct.

Source: O'Hagan & Stevens 2001

The prior influences a credible interval because the interval is derived from the posterior distribution, which combines the prior with the data; when the data are strong, the prior has little effect, but when the data are limited, the prior can substantially affect the interval. So the prior plays a role in determining a credible interval, especially with sparse data, which is why the choice of prior matters and its influence is often examined, since an informative prior can narrow or shift the interval, and reporting how the prior affects the credible interval supports transparent Bayesian inference where the interval reflects both the data and the prior beliefs.

Source: O'Hagan & Stevens 2001