Reliability as Projection in Operator-Theoretic Test Theory: Conditional Expectation, Hilbert Space Geometry, and Implications for Psychometric Practice
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Bibliographic record
Abstract
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" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="double-struck">E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>∣</mml:mo> <mml:mi mathvariant="script">G</mml:mi> <mml:mo stretchy="false">)</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" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Rel</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Var</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi mathvariant="double-struck">E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>∣</mml:mo> <mml:mi mathvariant="script">G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">/</mml:mo> <mml:mi mathvariant="normal">Var</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</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" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mspace width="0.25em"/> </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" display="inline" overflow="scroll"> <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.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.012 | 0.321 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.004 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it