From Linear Geometry to Nonlinear and Information-Geometric Settings in Test Theory: Bregman Projections as a Unifying Framework

This article develops a unified geometric framework linking expectation, regression, test theory, reliability, and item response theory through the concept of Bregman projection. Building on operator-theoretic and convex-analytic foundations, the framework extends the linear geometry of classical test theory (CTT) into nonlinear and information-geometric settings. Reliability and regression emerge as measures of projection efficiency-linear in Hilbert space and nonlinear under convex potentials. The exposition demonstrates that classical conditional expectation, least-squares regression, and information projections in exponential-family models share a common mathematical structure defined by Bregman divergence. By situating CTT within this broader geometric context, the article clarifies relationships between measurement, expectation, and statistical inference, providing a coherent foundation for nonlinear measurement and estimation in psychometrics.