Most clinicians and many medical statisticians interpret standard frequentist confidence intervals by invoking the Bayesian concept of subjective probability. Fortunately, the assumptions that render this interpretation acceptable are often quite reasonable in the setting of the practical day-to-day analysis of medical data. This article takes the subjective interpretation of confidence intervals to its logical conclusion and argues that the inferential understanding of clinicians and public health physicians could potentially be improved if, where it was appropriate, standard inferential statements--point estimates, 95 per cent confidence intervals and P-values--were supplemented by estimates of the subjective posterior probability, assuming a uniform prior density, that the true value of a parameter to be estimated exceeds one or a series of thresholds that are clinically critical or easily interpretable. Many decision makers in the health care arena draw totally inappropriate inferences from analyses where the point estimate indicates a clinically valuable effect but the null hypothesis cannot formally be rejected, and, although the proposed approach could be of potential value in a range of settings, it is argued that it could be of particular use in the rational interpretation of underpowered studies that must inform critical clinical or public health decisions.