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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>∣</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , representing the orthogonal projection of the observed score onto the σ-algebra of the latent variable. Reliability, expressed as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Rel</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>Var</mml:mi> <mml:mo>[</mml:mo> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>∣</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>]</mml:mo> <mml:mo>/</mml:mo> <mml:mi>Var</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> , quantifies the efficiency of this projection-the squared cosine between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi> <mml:mspace></mml:mspace></mml:mrow> </mml:math> and its true-score projection. This formulation unifies reliability with regression <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msup> </mml:mrow> </mml:math> , 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.