From 8 wins in 10 trades you report a 95% Wald confidence interval for the win rate and, separately, a 95% credible interval under a Beta(2, 2) prior. Compute both, and state precisely what each one claims.

From 8 wins in 10 trades you report a 95% Wald confidence interval for the win rate and, separately, a 95% credible interval under a Beta(2, 2) prior. Compute both, and state precisely what each one claims.

Approach: Build the normal-approximation interval from the sample proportion, then update the beta prior and take posterior quantiles, and compare the statements the two intervals make.

The Wald interval is 0.552 to 1.000 and the Beta(10, 4) posterior gives roughly 0.462 to 0.909, and only the second is a statement about where the parameter lies. The Wald interval is 0.8 plus or minus 1.96*sqrt(0.8*0.2/10) = 0.8 plus or minus 0.248, so it runs to 1.048 and has to be truncated at 1, which already shows the normal approximation failing at n = 10. Its claim is about coverage: over repeated samples, 95% of such intervals contain the fixed true p, and this particular interval either does or does not. The posterior is Beta(2 + 8, 2 + 2) = Beta(10, 4) with mean 10/14 = 0.714, and its central 95% region is a credible interval, whose claim is that given the prior and the data, p lies in it with probability 0.95. The credible interval is pulled below 0.8 because the prior adds one pseudo-win and one pseudo-loss.

Follow-up: Which prior makes the 95% credible interval for a binomial proportion coincide with the exact Clopper-Pearson interval, and does any prior make them agree at every observed count?

Key concepts: confidence interval, credible interval, conjugate prior, coverage.