Credible Interval Estimation

The Bayesian process of calculating a range of plausible values for a parameter based on its estimated posterior probability distribution.

Source: O'Hagan & Stevens 2001

Credible interval estimation is the Bayesian process of deriving a range of plausible values for a parameter from its posterior probability distribution, the distribution that results after combining prior belief with the data. It reads off the interval directly from that posterior, so a ninety-five per cent credible interval is simply the central range holding that much of the posterior probability. This yields the intuitive statement that the parameter probably lies within the range, which requires a prior to be specified alongside the data. Reading a range off the posterior is what it does. Sutton and Abrams (2001) describe this.

Source: Sutton & Abrams 2001

Credible interval estimation is carried out by first obtaining the posterior distribution of the parameter, combining the prior with the data through Bayes' theorem, and then finding a range of the posterior that contains the specified probability, such as the central interval between percentiles or the highest posterior density interval. Posterior samples from methods such as Markov chain Monte Carlo are often used. So credible interval estimation is carried out by deriving and summarising the posterior distribution, extracting an interval that holds the desired probability mass, which yields a range whose width reflects the posterior uncertainty about the parameter after the data and prior are combined.

Source: O'Hagan & Stevens 2001

Credible interval estimation derives a range from the posterior distribution and interprets it as containing the parameter with a stated posterior probability, while confidence interval estimation constructs a range from the sampling distribution and interprets it through long-run coverage over repeated samples. The Bayesian approach uses a prior and makes a direct probability statement about the parameter; the frequentist approach does not. So the two differ in framework and interpretation, with credible interval estimation treating the parameter probabilistically given the data and prior, and confidence interval estimation relying on the frequency properties of the procedure, reflecting the broader distinction between Bayesian and frequentist inference.

Source: O'Hagan & Stevens 2001

Credible interval estimation requires a prior distribution for the parameter, the data and a likelihood linking them to the parameter, and computation of the resulting posterior distribution, from which the interval is extracted. When the posterior has no closed form, computational methods such as Markov chain Monte Carlo are used to obtain posterior samples. So credible interval estimation needs a prior, a likelihood, and the means to compute the posterior, which distinguishes it from frequentist estimation by its use of a prior, and the quality of the resulting interval depends on the appropriateness of the prior and the model and on adequate computation of the posterior.

Source: O'Hagan & Stevens 2001

Credible interval estimation is used when a Bayesian approach is adopted, offering a direct probability statement about the parameter that many find more intuitive than the frequentist interpretation, and allowing prior information to be incorporated. It suits complex models and settings where prior knowledge is valuable. So credible interval estimation is used to express uncertainty about a parameter directly as a posterior probability, which is attractive for its interpretability and its ability to combine prior knowledge with data, and it is preferred where a Bayesian framework fits the problem, though it requires specifying a prior and computing the posterior.

Source: O'Hagan & Stevens 2001