From Linear Geometry to Nonlinear and Information-Geometric Settings in Test Theory: Bregman Projections as a Unifying Framework
Why this work is in the frame
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Bibliographic record
Abstract
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.
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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 | 0.007 | 0.219 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.007 |
| 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