Summary: We derive the differential equation that a prior must satisfy if the posterior probability of a one-sided credibility interval for a parametric function and its frequentist probability agree up to \(O (n^{-1})\). This equation turns out to be identical with Stein’s equation [C. Stein, Sequential Methods in Statistics, Banach Cent. Publ. 16, 485-514 (1985; Zbl 0662.62030)] for a slightly different problem, for which also our method provides a rigorous justification. Our method is different in details from Stein’s but similar in spirit to A. P. Dawid [J. R. Stat. Soc., Ser. B 53, No. 1, 79-109 (1991)] and P. J. Bickel and J. K. Ghosh [Ann. Stat. 18, No. 3, 1070-1090 (1990; Zbl 0727.62035)]. Some examples are provided.