Statistical & Clinical Analysis
Jeffreys Beta(0.5, 0.5) Binomial Posterior Mean & Standard Deviation Formula
Estimate rare-event and zero-numerator binomial proportions without collapsing to 0% using Harold Jeffreys’ invariant Beta(1/2, 1/2) posterior mean (x + 0.5)/(n + 1) and exact posterior standard deviation.
Why Jeffreys’ invariant Beta(1/2, 1/2) prior avoids both 0% Wald collapse and Laplace’s heavy +1/+2 pull!
When 0 adverse events occur in n = 18 Phase I patients or aerospace qualification tests, the naive MLE p_hat = x / n = 0% falsely implies zero risk with zero Wald standard error, while Laplace’s uniform Beta(1, 1) rule of succession (x + 1)/(n + 2) = 1/20 = 5.00% pulls rare-event rates aggressively toward 50%! Sir Harold Jeffreys (1946, Proceedings of the Royal Society A) derived the reparameterization-invariant Fisher-information prior Beta(1/2, 1/2) proportional to p^(-1/2) * (1 - p)^(-1/2)!
Under conjugate binomial updating, observing x successes in n trials yields exact posterior Beta(alpha = x + 0.5, beta = n - x + 0.5), with closed-form Posterior Mean = (x + 0.5) / (n + 1) and Posterior SD = SQRT(((x + 0.5) * (n - x + 0.5)) / ((n + 1)^2 * (n + 2)))! At x = 0, n = 18, Jeffreys’ posterior mean is 0.5 / 19 = 0.0263 (2.63%) with posterior SD = 0.0358—rising to 4.5 / 19 = 0.2368 when x = 4!
| Observed Binomial Trial (x, n) | Naive MLE x / n | Jeffreys Posterior Mean (x+0.5)/(n+1) | Jeffreys Posterior SD |
|---|---|---|---|
| x = 0 events in n = 18 trials | 0.0000 (0% collapse!) | 0.0263 (2.63% non-zero Bayes estimate) | 0.0358 |
| x = 4 events in n = 18 trials | 0.2222 (22.22%) | 0.2368 (mild 1/2-pseudo-count shrinkage) | 0.0951 |
| Invalid x = 22 > n = 18 | Impossible Count | "Invalid Binomial Counts" | 0 <= x <= n Guard |
Jeffreys Beta(x + 0.5, n - x + 0.5) Posterior Mean = (x + 0.5) / (n + 1)
Computes the conjugate Beta(1/2, 1/2) posterior mean for x events (B2) in n trials (B3).
=IF(OR(B2 < 0, B3 <= 0, B2 > B3), "Invalid Binomial Counts", ROUND((B2 + 0.5) / (B3 + 1), 4))Jeffreys Exact Beta Posterior Standard Deviation Formula
Evaluates SQRT((alpha * beta) / ((alpha + beta)^2 * (alpha + beta + 1))) where alpha = x + 0.5 and beta = n - x + 0.5.
=IF(OR(B2 < 0, B3 <= 0, B2 > B3), "Invalid Binomial Counts", ROUND(SQRT(((B2 + 0.5) * (B3 - B2 + 0.5)) / (((B3 + 1) ^ 2) * (B3 + 2))), 4))Try the example
Use the original values or change them to test your understanding.