We propose a novel Bayesian test under a (noninformative) Jeffreys'priorspecification. We check whether the fixed scalar value of the so-calledBayesian Score Statistic (BSS) under the null hypothesis is aplausiblerealization from its known and standardized distribution under thealternative. Unlike highest posterior density regions the BSS isinvariantto reparameterizations. The BSS equals the posterior expectation oftheclassical score statistic and it provides an exact test procedure,whereasclassical tests often rely on asymptotic results. Since the statisticisevaluated under the null hypothesis it provides the Bayesiancounterpart ofdiagnostic checking. This result extends the similarity of classicalsampling densities of maximum likelihood estimators and Bayesianposteriordistributions based on Jeffreys' priors, towards score statistics. Weillustrate the BSS as a diagnostic to test for misspecification inlinearand cointegration models.