Binomial Proportion Confidence Interval - Jeffreys Interval

Jeffreys Interval

The ‘Jeffreys interval’ has a Bayesian derivation, but it has good frequentist properties. In particular it has coverage properties that are similar to the Wilson interval, but it is one of the few intervals with the advantage of being ‘equal-tailed’ (e.g. for a 95% confidence interval, the probabilities of the interval lying above or below the true value are both close to 2.5%). In contrast, the Wilson interval has a systematic bias such that it is centred too close to p=0.5.

The Jeffreys interval is the Bayesian credible interval obtained when using the non-informative Jeffreys prior for the binomial proportion p. The Jeffreys prior for this problem is a Beta distribution with parameters (1/2, 1/2). After observing x successes in n trials, the posterior distribution for p is a Beta distribution with parameters (x + 1/2, n – x + 1/2).

When x ≠0 and x ≠ n, the Jeffreys interval is taken to be the 100(1 – α)% equal-tailed posterior probability interval, i.e., the α / 2 and 1 – α / 2 quantiles of a Beta distribution with parameters (x + 1/2, n – x + 1/2) . These quantiles need to be computed numerically, although this is reasonably simple with modern statistical software.

In order to avoid the coverage probability tending to zero when p → 0 or 1, when x = 0 the upper limit is calculated as before but the lower limit is set to 0, and when x = n the lower limit is calculated as before but the upper limit is set to 1.