Credible Interval - Contrasts With Confidence Interval

Contrasts With Confidence Interval

A frequentist 95% confidence interval of 35–45 means that with a large number of repeated samples, 95% of the calculated confidence intervals would include the true value of the parameter. The probability that the parameter is inside the given interval (say, 35–45) is either 0 or 1 (the non-random unknown parameter is either there or not). 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). Antelman (1997, p. 375) summarizes a confidence interval as "... one interval generated by a procedure that will give correct intervals 95 % of the time".

In general, Bayesian credible intervals do not coincide with frequentist confidence intervals for two reasons:

• credible intervals incorporate problem-specific contextual information from the prior distribution whereas confidence intervals are based only on the data;
• credible intervals and confidence intervals treat nuisance parameters in radically different ways.

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 will 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; and also if the unknown parameter is a scale parameter (i.e. the forward probability function has the form ), with a Jeffreys' prior — 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.