Meta-analysis as a barycenter of study distributions: information-geometric pooling, heterogeneity, and robustness

Meta-analysis usually reduces each study to an effect estimate with a standard error and pools these by inverse-variance weighting: fixed effect (FE), random effects (RE), or unrestricted weighted least squares (UWLS). We propose information-geometric meta-integration (IGMI), representing each study by its sampling distribution, the Gaussian N ( θ _i , Σ _i ), and pooling studies as a weighted Fréchet mean (barycenter) under Bures–Wasserstein (BW), Fisher–Rao, or Wasserstein–Fisher–Rao (WFR) geometry. In the scalar fixed-variance case the BW barycenter mean is exactly the FE estimate; the minimized Fréchet functional reproduces Higgins–Thompson I ² and DerSimonian–Laird and a Fréchet-scatter pivot reproduces the Hartung–Knapp–Sidik–Jonkman interval at m = 1 and yields an exact Hotelling F _{m,K − m} region for m outcomes under proportional total covariances. WFR adds a robust outlier-resistant pool: as its length scale δ → ∞ it converges monotonically to BW, whereas finite δ gives a redescending M-estimator with rejection point exactly πδ . Simulations show calibrated multivariate coverage at small K , where Wald intervals undercover, and strong resistance of the equal-weight WFR pool to contamination. In 2,445 Cochrane meta-analyses, WFR most often wins leave-one-out predictive scoring. In 835 bivariate meta-analyses, the closed-form BW barycenter matches REML multivariate meta-analysis predictively and is exactly invariant to the unreported within-study correlation, unlike the likelihood estimate. Methods are implemented in the development R package gtmeta , available as a GitHub source repository.