Reliability as Projection in Operator-Theoretic Test Theory: Conditional Expectation, Hilbert Space Geometry, and Implications for Psychometric Practice

This article reconceptualizes reliability as a theorem derived from the projection geometry of Hilbert space rather than an assumption of classical test theory. Within this framework, the true score is defined as the conditional expectation E ( X ∣ G ) , representing the orthogonal projection of the observed score onto the σ-algebra of the latent variable. Reliability, expressed as Rel ( X ) = Var [ E ( X ∣ G ) ] / Var ( X ) , quantifies the efficiency of this projection—the squared cosine between X and its true-score projection. This formulation unifies reliability with regression R 2 , factor-analytic communality, and predictive accuracy in stochastic models. The operator-theoretic perspective clarifies that measurement error corresponds to the orthogonal complement of the projection, and reliability reflects the alignment between observed and latent scores. Numerical examples and measure-theoretic proofs illustrate the framework’s generality. The approach provides a rigorous mathematical foundation for reliability, connecting psychometric theory with modern statistical and geometric analysis.