From Linear Geometry to Nonlinear and Information-Geometric Settings in Test Theory

Bregman Projections as a Unifying Framework

Bibliographic Data

ID20286409
AuthorsBruno D Zumbo (0000-0003-2885-5724, The University of British Columbia, Vancouver, Canada, corresponding author)
Year2025
Volume86
Issue4
Pages714-737
Publication date2025-12-12
Peer ReviewedYes
Open AccessYes
TypeARTICLE
VenueEducational and Psychological Measurement (JOURNAL)
Journal identifiersISSN: 0013-1644 • E-ISSN: 1552-3888
PublisherSAGE Publications (PUBLISHER • US)
DOI10.1177/00131644251393483
PMID41399675
OpenAlexW4417427433
LanguageEN
References cited26

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

Bregman divergence · Hilbert space · Information geometry · Nonlinear system · Projection (relational algebra) · Regular polygon · Reliability (semiconductor) · Space (punctuation) · Statistical hypothesis testing · Cognitive Abilities and Testing · Mental Health Research Topics · Psychometric Methodologies and Testing

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