When 2 + 2 = 5, 3, or 2: A Correlation-Aware Design Doctrine for Military Sensor Fusion

Nahin and Pokoski (1980) asked whether two plus two can equal five: under conditional independence, Bayesian fusion of several detectors can substantially exceed the best single one. This paper examines what survives when the detectors’ errors are correlated. Working in posterior-odds / log-likelihood-ratio form, it shows that with a known correlation matrix R the fused detection index is governed by the effective sensor count K_eff = 1ᵀR⁻¹1 (equal to K/[1+(K−1)ρ] for equicorrelation). A fusion rule that assumes independence is overconfident by exactly 1+(K−1)ρ and, under asymmetric cost, misplaces the decision threshold, incurring excess Bayes risk that grows with correlation. With the covariance known, the inverse-covariance rule removes both faults, and adding a detector never lowers the optimal index; the “2+2<4” pathology belongs to the naive equal-weight rule. The paper is explicit about the limits of these known-covariance results. Correlation information is scale-free; the split of observed covariance into common and independent error is not identifiable from data alone (the Frisch problem, Wilson’s inequality); and the optimal weights solve an ill-posed inverse problem that neither plug-in inversion nor regularisation resolves without information about the errors stated explicitly. With estimated covariances, adding sensors can reduce performance. The result is a design doctrine for military sensor fusion. Files: the paper (PDF) and a reproducible package (LaTeX source, figures, and three short Python scripts that regenerate every number and figure). This release discloses the foundations; practical extensions are unpublished and available for discussion on request.