Bayesian Inference for Incomplete 2 × 2 Diagnostic Tables

Diagnostic accuracy studies often report only part of the 2×2 table, leaving some of the key denominators or cell counts unknown and preventing direct calculation of sensitivity, specificity, and related measures. In this paper, we develop Bayesian methods for reconstructing such incomplete tables when only the test-positive row is reported, and when the total sample size N is either unknown or known. The reported counts restrict the missing cells to a set of compatible tables but cannot fully distinguish among them, so prior assumptions can strongly influence the reconstruction. We propose a rule for eliciting accuracy priors from published estimates and show how reported summaries, such as rounded sensitivity, specificity, or accuracy, can be incorporated as hard or soft compatibility constraints. Posterior inference uses direct summation over the discrete table space, with numerical integration over continuous hyperparameters where required. In a simulation study, reconstructions from the counts alone were either highly uncertain or, under strongly directional accuracy priors combined with a dispersed negative-binomial denominator prior, precise but frequently incorrect, whereas adding a single rounded sensitivity computed from the true table yielded narrow credible intervals that contained the true value in all settings; when the reported sensitivity was perturbed, coverage fell unless the compatibility tolerance was widened. The methods are illustrated using a benchmark breast magnetic resonance imaging (MRI) table and a published study of artificial intelligence-based detection of intracranial hemorrhage. The output is a posterior distribution over compatible tables and should not be used as observed data in subsequent analyses.