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A gravitational-wave background can be detected in pulsar-timing-array dataas Hellings--Downs correlations among the timing residuals measured fordifferent pulsars. The optimal statistic implements this concept as a classicalnull-hypothesis statistical test: a null model with no correlations can berejected if the observed value of the statistic is very unlikely under thatmodel. To address the dependence of the statistic on the uncertain pulsar noiseparameters, the pulsar-timing-array community has adopted a hybridclassical--Bayesian scheme (Vigeland et al. 2018) in which the posteriordistribution of the noise parameters induces a posterior distribution for thestatistic. In this article we propose a rigorous interpretation of the hybridscheme as an instance of posterior predictive checking, and we introduce a newsummary statistic (the Bayesian signal-to-noise ratio) that should be used toaccurately quantify the statistical significance of an observation instead ofthe mean posterior signal-to-noise ratio, which does not support such a directinterpretation. In addition to falsifying the no-correlation hypothesis, theBayesian signal-to-noise ratio can also provide evidence supporting thepresence of Hellings--Downs correlations. We demonstrate our proposal withsimulated datasets based on NANOGrav's 12.5-yr data release. We also establisha relation between the posterior distribution of the statistic and the Bayesfactor in favor of correlations, thus calibrating the Bayes factor in terms ofhypothesis-testing significance.

Posterior predictive checking for gravitational-wave detection with pulsar timing arrays: I. The optimal statistic