From Linear Geometry to Nonlinear and Information-Geometric Settings in Test Theory: Bregman Projections as a Unifying Framework

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.

Building on foundational contributions from Zimmerman (1975) and Zimmerman and Zumbo (2001), Zumbo (in press-a) recently described an operator-theoretic framework reconceptualizing reliability as a theorem derived from the projection geometry of Hilbert space rather than as an assumption of classical test theory (CTT). This paper extends the linear operator-theoretic framework developed in the companion work (Zumbo, in press-a) to nonlinear and information-geometric settings.

In that earlier formulation, projection and reliability were defined within the Euclidean geometry of $L^2$ . Here, those concepts are generalized through Bregman divergences (see, for example, Banerjee & Basu, 2017; Bregman, 1967), allowing projections to occur in curved spaces defined by convex potentials. This extension links regression, expectation, and reliability within a single geometric principle: each represents the projection of observed data onto a subspace that captures systematic variation. In psychometric terms, the subspace corresponds to the latent structure underlying observed responses. Thus, classical reliability and regression appear as special cases of Bregman projection—linear under the quadratic potential and nonlinear under information-theoretic geometries.

Motivation and Applied Implications

Psychometric practice increasingly involves complex data involving nonlinear relationships as a result of, for example, ordinal items responses, bounded scales, and skewed distributions (Raykov & Marcoulides, 2011). Existing extensions (e.g., item response theory [IRT], generalized linear latent models) handle these through specialized link functions, but often lack a unifying theoretical foundation. The Bregman framework addresses this need by providing a single geometric principle—projection under a convex potential—that governs both linear and nonlinear models.

In applied test analysis, this generalization allows reliability, regression, and scoring procedures to be interpreted consistently across data types and model forms. For example, reliability defined via divergence rather than variance remains meaningful for ordinal, categorical, or information-based measures, preserving the projection interpretation of true score even outside Euclidean spaces. This perspective reframes model comparison, measurement precision, and estimation efficiency as geometric questions about curvature and projection alignment, thereby offering a unified theoretical and applied toolkit for modern psychometrics.

This geometric perspective transforms measurement from a question of linear decomposition to one of projection alignment within curved spaces, offering a principled foundation for both theoretical innovation and applied psychometric practice.

Aim and Overview

As noted by Zumbo (in press-a), nearly six decades have passed since Zimmerman and his collaborators (e.g., Williams & Zimmerman, 1977; Zimmerman, 1969a, 1969b, 1969c, 1970, 1972, 1975, 1976; Zimmerman et al., 1968) established a rigorous, probabilistic, and geometric foundation for CTT, much of it published in this journal. The current line of inquiry revisits, extends, and clarifies these foundational ideas for a new generation of psychometricians. Building on that legacy, this paper extends the operator-theoretic approach (Zumbo, in press-a) to nonlinear and information-geometric settings, showing that both operator-theoretic and Bregman formulations share a common structure: reliability and expectation as projection operators—linear in Hilbert space, nonlinear under convex potentials. Applications to IRT reveal that expected item response functions constitute Bregman projections of observed responses onto curved information manifolds parameterized by latent traits. The exposition emphasizes conceptual insight and accessibility for applied researchers.

The remainder of this paper is organized as follows. The next section revisits projection and reliability within the Euclidean geometry of $L^2$ , providing the operator-theoretic foundations upon which the nonlinear extension is built. Following this, the paper introduces Bregman projections and develops their analytic and convex foundations, clarifying how curvature and dual geometry generalize classical orthogonal projection. Subsequent sections formalize the concept of generalized conditional expectation, Bregman regression, and reliability as projection efficiency under convex divergence, demonstrating their continuity with the linear operator-theoretic framework. The exposition then applies this unified geometric formulation to psychometric test theory and IRT, showing that expected item response functions can be interpreted as Bregman projections on curved information manifolds. Later sections examine the operator-theoretic properties, monotonicity, and mathematical coherence of the generalized reliability measure, and compare the Bregman framework with other geometric generalizations such as kernel-based and Bayesian approaches. The paper concludes by situating these results within a broader geometric synthesis of measurement, expectation, and reliability, and by discussing implications for nonlinear, information-geometric approaches to psychometric modeling.

Preliminaries: Projection and Reliability Within the Euclidean Geometry of L²

As described by Zumbo (in press-a), let $(\Omega ,\mathcal{F},P)$ be a probability space, and let $X\in L^2(\Omega ,\mathcal{F},P)$ be the observed score. For a sub- $\mathrm{\sigma }$ -algebra $\mathcal{G}\subseteq \mathcal{F}$ , the true score of $X$ with respect to $\mathcal{G}$ is the conditional expectation

$$T=\mathbb{E}[X|\mathcal{G}],$$

which coincides with the orthogonal projection

$$T=P_{\mathcal{G}}X,$$

where $P_{\mathcal{G}}$ is the projection operator from $L^2(\Omega ,\mathcal{F},P)$ onto $L^2(\Omega ,\mathcal{G},P)$ that maps $X\in L^2(\Omega ,\mathcal{F},P)$ to a $\mathcal{G}$ -measurable random variable. Thus, $\mathcal{G}$ defines the information set, and the conditional expectation isolates the systematic component of the observed score. That is, in operator-theoretic formulations, $\mathcal{G}$ generates the true-score subspace of $L^2(\Omega ,\mathcal{F},P)$ . From a psychometric perspective, $\mathcal{G}$ encodes the relevant sources of systematic variation. By itself, $\mathcal{G}$ is not an operator but a collection of measurable events representing person-, situation-, ecological-, or contextual-level factors relevant to the construct (Zumbo, 2023).

Reliability, defined as the variance ratio

$$\mathrm{Rel}(X)=\frac{\mathrm{Var}\left[\mathbb{E}(X|\mathcal{G})\right]}{\mathrm{Var}\left(X\right)},$$

quantifies projection efficiency—the squared cosine between observed and true-score projections. This operator-theoretic view unifies reliability, regression $R^2$ , and predictive accuracy in stochastic models within a shared geometric structure.

Building on this foundation, Zumbo (in press-b) extended the theory using the Rayleigh quotient to define polychoric and copula-based estimators for composite reliability. These developments positioned theta reliability (Armor, 1974) as a robust, theoretically grounded alternative to Cronbach’s alpha (Cronbach, 1951) and motivated a broader generalization of reliability and regression through Bregman projections and non-Euclidean geometries.

The sections that follow not only formalize the Bregman projection but also contrast it with other generalizations of Euclidean test geometry, clarifying its distinct advantages for modern psychometric modeling.

Bregman Projections, Statistics, and Probability

Analytic and Convex Foundations

Functional analysis provides the linear framework in which expectation and reliability emerge as orthogonal projections in Hilbert space. Convex analysis extends this structure by replacing linearity with curvature, allowing projection and optimization to occur on non-Euclidean manifolds. The Bregman divergence arises from this shift, measuring discrepancy through the tangent geometry induced by a convex potential $\varphi$ . Thus, convex analysis provides the mathematical foundation for understanding nonlinearity and information geometry within measurement and inference.

Together, functional and convex analysis reveal that linear and nonlinear measurement models share a common geometric foundation: projection as optimal approximation under either squared distance or convex divergence.

Although no single monograph focuses exclusively on Bregman projections in statistics and probability, their connection to these fields is well established across research papers, conference proceedings, and specialized texts in convex analysis, optimization, and statistical inference. Bregman projections play key roles in statistical estimation, robust statistics, and optimal transport, where they are closely related to algorithms such as the iterative proportional fitting procedure. Readers seeking a detailed discussion of the underlying mathematics may consult (Amari, 2016; Amari et al., 2018; Banerjee et al., 2005a, 2005b; Bauschke & Combettes, 2017; Bregman, 1967; Calin & Udrişte, 2014; Correa et al., 2023; Dodds et al., 1990; Li, 2021; Nielsen, 2019; Nielsen et al., 2022; Nielsen & Nock, 2014; Stummer & Vajda, 2012). Collectively, these works provide some of the mathematical foundations for the geometric and operator-theoretic concepts discussed in this paper.

Nonlinearity of Bregman Projections

This subsection clarifies the nonlinearity of Bregman projections, explains its geometric origin, and situates the concept within the broader framework developed in this paper.

Although Bregman projections generalize the classical notion of orthogonal projection, they are generally nonlinear operators. In the Euclidean setting, the projection of a random variable or vector $X$ onto a closed subspace $S\subset L^2$ is linear because it satisfies the superposition principle: for all scalars $a,b\in \mathbb{R},$

$$P_S(a{X}_1+b{X}_2)=a{P}_S(X_1)+b{P}_S(X_2).$$

This linearity arises from the quadratic potential $\varphi (x)=\frac{1}{2}{x}^2$ , which induces a flat Euclidean geometry governed by the Pythagorean theorem.

When $\varphi$ is non-quadratic, the geometry becomes non-Euclidean, and the corresponding Bregman projection,

$$P_C^{\varphi }(x)=\arg \underset{y\in C}{\min }{D}_{\varphi }(y,x),$$

depends on the local curvature of the potential. Because the Bregman divergence $D_{\varphi }(y,x)$ is asymmetric and does not satisfy the parallelogram law, the mapping $x\to P_C^{\varphi }(x)$ cannot, in general, be represented as a linear transformation.

Geometrically, the convex potential $\varphi$ defines dual coordinate systems: the primal space, parameterized by $x$ , and the dual space, parameterized by its gradient $\nabla \varphi (x)$ . In the dual space, the projection becomes linear:

$$\nabla \varphi (P_C^{\varphi }(x))=P_{\nabla \varphi (C)}(\nabla \varphi (x)),$$

where $P_{\nabla \varphi (C)}$ denotes the ordinary Euclidean projection in dual coordinates. However, when mapped back to the primal space through the inverse transformation $(\nabla \varphi {)}^{-1}$ , the operation becomes nonlinear. Consequently, Bregman projection is linear in the dual geometry but nonlinear in the original coordinates. This duality illustrates the principle that linearity in Euclidean geometry corresponds to affinity in information geometry, where the convex potential introduces curvature.

When $\varphi (x)=\frac{1}{2}{x}^2$ , the gradient $\nabla \varphi (x)=x$ , so the primal and dual coordinates coincide. The Bregman projection then reduces to the familiar orthogonal projection in $L^2$ , which is linear whenever the projection set $C$ is a subspace. For all other convex potentials, however, the Bregman projection remains nonlinear in the primal space, capturing the curvature of the underlying statistical manifold.

This geometric distinction shows that Bregman projections extend—rather than merely generalize—Euclidean projections. They describe projection in curved spaces where expectation, regression, and reliability are governed by divergence rather than distance.

What Is a Bregman Projection?

A Bregman projection extends the concept of orthogonal projection by replacing Euclidean distance with a Bregman divergence. Given a strictly convex and differentiable function $\varphi$ , the Bregman divergence between two points $x$ and $y$ is defined as

$$D_{\varphi }(x,y)=\varphi (x)-\varphi (y)-〈\nabla \varphi (y),x-y〉.$$

This divergence induces a geometric structure in which distances and projections behave differently from those in standard Euclidean space.

Dual Geometry and the Primal Space

The primal space refers to the original coordinate system in which data or variables naturally reside. The dual space, by contrast, is defined through the gradient map $\mathbf{\nabla }\mathbf{\varphi }$ , which transforms points from the primal space into coordinates of the dual geometry. This dual representation often possesses simpler geometric properties, particularly linearity, making it advantageous for both analysis and computation.

Linearity in the Dual Space

In the dual space, the Bregman projection becomes a linear operation because the divergence behaves analogously to a squared Euclidean distance. Consequently, projection onto a convex set—such as a hyperplane or affine subspace—can be computed using standard linear algebraic tools. This property parallels the operation of the mirror descent algorithm, in which updates are linear in the dual space but become nonlinear when transformed back to the primal space.

Nonlinearity in the Primal Space

When the projected point is mapped back to the original (primal) space using the inverse gradient $(\mathbf{\nabla }\mathbf{\varphi }{)}^{\mathbf{-}1}$ , the transformation is generally nonlinear. This nonlinearity arises because the gradient map $\mathbf{\nabla }\mathbf{\varphi }$ itself is nonlinear unless $\mathbf{\varphi }$ is quadratic. Thus, while projections are linear in the dual space, their images in the primal space often appear curved or distorted, reflecting the underlying geometry defined by the convex potential $\mathbf{\varphi }$ .

Bregman Divergence and Generalized Expectation

Let $\varphi :\mathbb{R}\to \mathbb{R}$ be a differentiable, strictly convex function, often called a convex potential. The Bregman divergence associated with $\varphi$ is defined for any two points $y,\hat{y}\in \mathbb{R}$ as

$$D_{\varphi }(y,\hat{y})=\varphi (y)-\varphi (\hat{y})-{\varphi }^{\text{'}}(\hat{y})(y-\hat{y}).$$

This quantity measures the excess of $\varphi (y)$ over its first-order (tangent-line) approximation at $\hat{y}$ . The divergence is always nonnegative and equals zero only when $y=\hat{y}$ . Unlike the Euclidean distance, it is generally asymmetric—that is, $D_{\varphi }(y,\hat{y})\ne D_{\varphi }(\hat{y},y)$ .

When $\varphi (x)=\frac{1}{2}{x}^2$ , the divergence simplifies to the squared Euclidean distance,

$$D_{\varphi }(y,\hat{y})=\frac{1}{2}(y-\hat{y}{)}^2,$$

which underlies least-squares estimation and the Hilbert-space projection operator.

For any closed convex set $C\subset {\mathbb{R}}^n$ , the Bregman projection of a point $x$ onto $C$ is

$$P_C^{\varphi }(x)=\arg \underset{y\in C}{\min }{D}_{\varphi }(y,x),$$

where the minimizer exists and is unique because $D_{\varphi }$ is convex in its first argument. This operator extends the orthogonal projection and satisfies the Bregman Pythagorean identity,

$$D_{\varphi }(z,x)=D_{\varphi }(z,P_C^{\varphi }(x))+D_{\varphi }(P_C^{\varphi }(x),x),z\in C,$$

which extends the familiar Pythagorean theorem from Euclidean to Bregman geometry.

In probability and statistics, the conditional expectation $\mathbb{E}[X\mid \mathcal{G}]$ is the orthogonal projection of $X$ onto the closed subspace $L^2(\mathcal{G})$ within $L^2(\mathcal{F})$ . This subspace represents the component of the measurement process determined by construct-relevant attributes (Zumbo, in press-a). Formally, $\mathbb{E}[X\mid \mathcal{G}]$ is the unique $\mathcal{G}$ -measurable random variable minimizing

$$\mathbb{E}[(X-Y{)}^2],Y\in L^2(\mathcal{G}).$$

This characterization extends directly to Bregman geometry. For any convex potential $\varphi$ , define the generalized conditional expectation as

$$E_{\varphi }[X\mid \mathcal{G}]=\arg \underset{Y\in L^1(\mathcal{G})}{\min }\mathbb{E}[D_{\varphi }(Y,X)].$$

This operator represents the Bregman projection of $X$ onto the subspace of $\mathcal{G}$ -measurable functions. When $\varphi (x)=\frac{1}{2}{x}^2$ , this definition reduces to the classical conditional expectation,

$$E_2[X\mid \mathcal{G}]=\mathbb{E}[X\mid \mathcal{G}].$$

Thus, $E_2$ is a special case of $E_{\varphi }$ associated with the Euclidean geometry of $L^2$ . The generalized projection framework unifies conditional expectation, regression, and reliability as solutions to divergence-minimization problems. It also accommodates non-quadratic loss functions, supporting robust, entropy-based, and information-theoretic approaches to measurement and prediction.

Bregman Regression

As shown by Zumbo (in press-a), reliability, regression $R^2$ , and factor communality all represent projection efficiency within different subspaces of $L^2$ , linking observed and latent variables through a common geometric principle. Measurement error is simply the orthogonal complement of this projection, capturing the variance in $X$ unaligned with the latent variable. As such, classical regression can be viewed as a projection problem within Euclidean geometry. In least-squares regression, the goal is to find a function $f(X)$ that minimizes expected squared loss,

$$\hat{f}=\arg \underset{f\in \mathcal{H}}{\min }\mathbb{E}[(Y-f(X){)}^2],$$

which represents the orthogonal projection of $Y$ onto the subspace of functions of $X$ in $L^2$ .

The Bregman generalization replaces the quadratic loss with a divergence determined by a convex potential $\varphi$ :

$${\hat{f}}_{\varphi }=\arg \underset{f\in \mathcal{H}}{\min }\mathbb{E}[D_{\varphi }(Y,f(X))].$$

The resulting estimator is the Bregman projection of $Y$ onto the model space $\mathcal{H}$ . When $\varphi (x)=\frac{1}{2}{x}^2$ , this formulation reduces to least squares; for other potentials, it produces estimation procedures corresponding to different statistical models.

The first-order optimality condition for Bregman regression is

$$\mathbb{E}[{\varphi }^{\text{'}}(Y)\mid X]={\varphi }^{\text{'}}({\hat{f}}_{\varphi }(X)).$$

This condition generalizes the normal equations of least squares, showing that the derivative of the potential ${\varphi }^{\text{'}}(Y)$ determines how data are projected in the transformed space defined by $\varphi$ . When $\varphi$ is quadratic, this condition becomes $\mathbb{E}[Y\mid X]=\hat{f}(X)$ ; when $\varphi (x)=x\log x-x$ , it yields the moment condition for Poisson regression; and when

$$\varphi (x)=x\log x+(1-x)\log (1-x),$$

it produces the cross-entropy condition for logistic regression.

From this perspective, many common estimation methods—including least-squares, Poisson, Gamma, and logistic regression—are special cases of Bregman regression, each arising from a specific potential function. The geometry of the potential defines both the loss function and the projection structure of the model.

The geometric logic of Bregman regression extends naturally to psychometric test theory. In both contexts, estimation involves projecting observed data onto a structured subspace that represents systematic variation—predictors in regression or latent traits in measurement models. This shared structure allows reliability, traditionally defined in terms of variance, to be reformulated as a measure of projection efficiency under the same divergence geometry. Thus, test theory can be viewed not as a separate statistical domain but as an applied instance of Bregman projection, where the goal is to quantify how effectively observed responses align with their latent expectations.

Bregman Projection and Psychometric Test Theory

It bears worth restating that in test theory, the true score is defined as the projection

$$T=\mathbb{E}(X\mid \mathcal{G}),$$

and reliability quantifies the proportion of variance explained by this projection:

$$\mathrm{Rel}(X)=\frac{\mathrm{Var}[\mathbb{E}(X\mid \mathcal{G})]}{\mathrm{Var}(X)}.$$

Within Bregman geometry, the true score generalizes to

$$T_{\varphi }=E_{\varphi }[X\mid \mathcal{G}]=\arg \underset{Y\in L^1(\mathcal{G})}{\min }\mathbb{E}[D_{\varphi }(Y,X)],$$

and reliability becomes

$$\mathrm{Re}{\mathrm{l}}_{\varphi }(X)=1-\frac{\mathbb{E}[D_{\varphi }(X,T_{\varphi })]}{\mathbb{E}[D_{\varphi }(X,\mathbb{E}[X])]}.$$

When $\varphi (x)=\frac{1}{2}{x}^2$ , this expression reduces to the classical variance ratio. For other convex potentials, it yields reliability measures that reflect information-theoretic or entropy-based geometries. This generalization extends the concept of reliability to complex data-generating processes that induce nonlinear relationships through ordinal responses, bounded measurement scales, or asymmetric (skewed) distributions.

By reframing both regression and reliability as Bregman projections, psychometric theory gains a unified geometric foundation. The same principle that governs estimation in generalized linear models also defines reliability as projection efficiency. This synthesis links traditional psychometric constructs—true scores, regression, and reliability coefficients—to modern convex and information-geometric methods, providing a rigorous and flexible framework for theoretical and applied measurement.

In this view, reliability is not tied to a specific metric. However, it represents a universal measure of projection efficiency—how effectively a given geometry (quadratic, power-law, or informational) captures systematic variation in observed data.

From an applied perspective, this geometric reformulation directly informs applied test analysis. It provides a unified lens for interpreting traditional reliability coefficients, item response models, and nonlinear estimation methods as instances of the same projection principle. By identifying the underlying geometry of each model, researchers can evaluate how measurement precision, model fit, and information efficiency emerge from a single mathematical operation—projection under a convex potential.

Bregman Projection and Operator-Theoretic Test Theory

An operator-theoretic test theory formalizes measurement and reliability as projection phenomena in Hilbert space (Zumbo’s in press-a). The conditional expectation operator

$$E_2:L^2(\mathcal{F})\to L^2(\mathcal{G}),E_2(X)=\mathbb{E}[X\mid \mathcal{G}],$$

projects an observed score $X$ onto the subspace of $\mathcal{G}$ -measurable random variables. In this setting, reliability represents the efficiency of that projection,

$$\mathrm{Rel}(X)=\frac{\mathrm{Var}[E_2(X)]}{\mathrm{Var}(X)}.$$

The corresponding orthogonal decomposition,

$$X=E_2(X)+(X-E_2(X)),$$

defines the true score and error components of CTT. The operator $E_2$ is linear, self-adjoint, and idempotent, reflecting the flat geometry of $L^2$ , where projection minimizes squared distance.

The Bregman framework generalizes this structure to convex geometries. Instead of the quadratic potential defining Euclidean space, it employs a differentiable, strictly convex function $\varphi$ that induces the Bregman divergence,

$$D_{\varphi }(y,x)=\varphi (y)-\varphi (x)-{\varphi }^{\text{'}}(x)(y-x),$$

and defines the generalized conditional expectation as

$$E_{\varphi }[X\mid \mathcal{G}]=\arg \underset{Y\in L^1(\mathcal{G})}{\min }\mathbb{E}[D_{\varphi }(Y,X)].$$

When $\varphi (x)=\frac{1}{2}{x}^2$ , this operator reduces to $E_2$ ; for other convex potentials, it becomes a nonlinear mapping that preserves the projection principle within the curved geometry determined by $\varphi$ .

Although $E_{\varphi }$ is nonlinear, it retains the essential properties of its Hilbert-space counterpart—idempotence, minimality, and generalized orthogonality under divergence geometry. In both frameworks, projection identifies the element of the latent subspace that best approximates the observed score, minimizing expected loss—whether defined by variance or divergence.

Mathematical Clarifications and Operator Properties

Although full proofs are beyond the intended aim and purpose of this article, the main operator-theoretic results can be summarized.

The generalized conditional expectation $E_{\varphi }[X\mid \mathcal{G}]$ is defined by

$$E_{\varphi }[X\mid \mathcal{G}]=\arg \underset{Y\in L^1(\mathcal{G})}{\min }\mathbb{E}[D_{\varphi }(Y,X)]$$

inherits several core properties of the Hilbert-space projection $E_2$ . Specifically, it satisfies:

1. Idempotence: $E_{\varphi }[E_{\varphi }[X\mid \mathcal{G}]\mid \mathcal{G}]=E_{\varphi }[X\mid \mathcal{G}]$ , ensuring internal consistency of the projection operator.

2. Minimality: The image of $E_{\varphi }$ minimizes expected divergence within the $\mathcal{G}$ -measurable subspace, generalizing least-squares optimality.

3. Generalized Orthogonality: The residual $X-E_{\varphi }[X\mid \mathcal{G}]$ is orthogonal to the subspace in the sense that

$$\mathbb{E}[〈\nabla \varphi (X)-\nabla \varphi (E_{\varphi }[X\mid \mathcal{G}]),Y-E_{\varphi }[X\mid \mathcal{G}]〉]=0$$

for all $Y\in L^1(\mathcal{G})$ .

These results are well established in convex-analytic and information-geometric literature (see Amari, 2016; Banerjee et al., 2005a; Nielsen & Nock, 2014). Readers interested in formal derivations may consult Appendix A , where the proofs and operator conditions are sketched for completeness.

Monotonicity Properties of Rel $\mathrm{Re}{\mathrm{l}}_{\varphi }(X)$

A natural question concerns whether the generalized reliability measure

$$\mathrm{Re}{\mathrm{l}}_{\varphi }(X)=1-\frac{\mathbb{E}[D_{\varphi }(X,T_{\varphi })]}{\mathbb{E}[D_{\varphi }(X,\mathbb{E}[X])]}$$

preserves the monotonicity and boundedness properties of the classical reliability coefficient.

Under standard regularity conditions—strict convexity of $\varphi$ , non-degeneracy of $X$ , and well-defined expectations— $\mathrm{Re}{\mathrm{l}}_{\varphi }(X)$ satisfies:

1. Range constraint: $0\le \mathrm{Re}{\mathrm{l}}_{\varphi }(X)\le 1$ , since the Bregman projection minimizes expected divergence.

2. Monotonicity: For nested sub- $\sigma$ -fields ${\mathcal{G}}_1\subseteq {\mathcal{G}}_2,$

$${\mathrm{Rel}}_{\varphi }^{({\mathcal{G}}_1)}(X)\le {\mathrm{Rel}}_{\varphi }^{({\mathcal{G}}_2)}(X),$$

ensuring that finer latent representations cannot reduce reliability.

• 3. Consistency: When $\varphi (x)=\frac{1}{2}{x}^2$ , these properties reduce to those of the standard variance-based reliability.

These results are established based on Banerjee and Basu’s (2017) description of generalized conditional expectations in Bregman geometry, Nielsen and Boltz (2011), and information-geometric literature (see Amari & Nagaoka, 2000). The monotonicity proof follows directly from the Pythagorean property of Bregman divergences, which ensures that any additional conditioning information (i.e., expanding $\mathcal{G}$ ) can only decrease expected divergence and thus increase reliability. A brief demonstration of this result is outlined in Appendix B .

Comparison With Other Geometric Generalizations

Several frameworks extend CTT beyond Euclidean geometry, including kernel-based Hilbert spaces and Bayesian geometric approaches. Kernel methods generalize linear models through implicit feature mappings, increasing representational flexibility while preserving an underlying Hilbertian (inner product) structure.

In contrast, Bregman geometry replaces the inner product with a convex potential, creating a family of non-Euclidean geometries that generalize both linear and exponential families. This substitution shifts the foundation of measurement from distance to divergence, enabling projection and estimation to be defined by minimizing expected convex loss rather than squared error.

Bayesian geometric frameworks, by comparison, represent uncertainty through probability manifolds and the Fisher–Rao metric, emphasizing curvature in posterior distributions. Although they capture probabilistic structure, they operate within fixed priors and parameterizations. The Bregman framework differs by generalizing the loss function itself, creating a direct link between estimation, reliability, and information-theoretic efficiency without requiring prior distributions or parametric assumptions.

Thus, Bregman geometry unifies loss-based optimization and geometric representation within a single convex-analytic framework. It encompasses Euclidean distance, cross-entropy, and Kullback–Leibler (KL) divergence as special cases, extending both kernel and Bayesian generalizations by redefining measurement geometry at the level of divergence rather than representation.

In this unified perspective, both theories articulate reliability as projection efficiency—the degree to which observed scores align with their latent projections. The operator-theoretic formulation situates this principle within the spectral structure of linear operators in $L^2$ ; the Bregman formulation situates it within the curvature and duality of convex potentials. Together, they describe a continuous theoretical lineage—from classical Euclidean test theory to modern nonlinear, information-geometric measurement models.

Thus, the operator-theoretic and Bregman approaches express the same principle: measurement as projection from an observed space onto a latent structure. The operator-theoretic model describes the linear Euclidean case, while the Bregman framework extends it to nonlinear, information-geometric spaces. The conditional expectation operator $E_2$ is the Euclidean instance of the more general $E_{\varphi }$ , just as least-squares regression is the Euclidean instance of Bregman regression. The shift from squared error to general divergence transforms projection from a linear to a convex operation, allowing measurement and reliability to be understood in information-geometric as well as algebraic terms.

In this unified perspective, both theories articulate reliability as projection efficiency—the degree to which observed scores align with their latent projections. The operator-theoretic formulation situates this principle within the spectral structure of linear operators in $L^2$ ; the Bregman formulation situates it within the curvature and duality of convex potentials. Together, they describe a continuous theoretical lineage—from classical, Euclidean test theory to modern, nonlinear, information-geometric measurement models.

Having situated psychometric reliability within this broader Bregman framework, the next step is to formalize the operator-theoretic properties that ensure the generalization remains mathematically coherent. The following section develops the underlying geometric foundations—idempotence, minimality, and generalized orthogonality—that parallel the properties of linear projection in $L^2$ .

These results establish that the nonlinear Bregman operator preserves the essential structure of expectation and reliability, even when the measurement geometry departs from Euclidean form. They provide the rigorous bridge between the conceptual interpretation of reliability as projection efficiency and its formal realization in curved, information-geometric spaces.

Geometric Foundations of Psychometric Theory

In test theory, the projection of observed scores onto their latent expectations occurs in a Euclidean geometry defined by the quadratic potential $\varphi (x)=\frac{1}{2}{x}^2$ .

Models such as IRT, generalized linear latent models, and ordinal reliability extend this structure to non-Euclidean geometries induced by other convex potentials.

In these settings, the mapping between latent and observed scores becomes nonlinear, and reliability reflects projection efficiency under a Bregman divergence rather than variance reduction alone.

This unified geometric framework situates both linear and nonlinear measurement models within a single theoretical continuum, linking regression, reliability, and estimation as forms of projection in curved statistical spaces.

Both regression and IRT can be viewed as Bregman projections. In linear regression, fitted values are the orthogonal projection of outcomes onto the predictor subspace, minimizing squared error—the divergence generated by the quadratic potential. In IRT, estimation occurs in a different geometry: the potential corresponds to the log-likelihood or cross-entropy, and projection takes place in the curved manifold defined by the model’s link function. Despite their apparent differences, both frameworks perform the same operation—identifying the element in a constrained space that minimizes expected divergence from observed data. Thus, regression, reliability, and IRT all express a single geometric principle of approximation under different convex potentials.

Nonlinearity in Psychometric Models

Understanding that reliability, regression, and IRT share this projection principle clarifies the role of nonlinearity in psychometric theory. In Euclidean geometry, projections are linear, producing additive and proportional relationships between observed and latent variables.

In the broader Bregman setting, the geometry is curved, and projections become nonlinear when expressed in the original measurement space. This curvature introduces the multiplicative and logistic relations that characterize modern psychometric models.

CTT assumes linear, additive relationships—a direct consequence of the quadratic potential.

However, when measurement processes follow nonlinear or multiplicative patterns, the Bregman framework provides a more general curved geometric description. For example, in logistic or Poisson models, projections occur along manifolds defined by the link function of $\varphi$ . These nonlinear projections preserve the idea of “best approximation,” but redefine optimality in terms of divergence rather than variance reduction.

In Euclidean space, reliability quantifies the proportion of variance explained by a linear projection. Under non-Euclidean geometries, reliability quantifies the reduction in expected divergence between observed and latent scores. The generalized measure, denoted $\mathrm{Re}{\mathrm{l}}_{\varphi }(X)$ , adapts reliability to the curvature of the measurement space. This interpretation aligns naturally with models such as logistic IRT and generalized linear latent frameworks, where observed–latent relations are inherently nonlinear. Thus, the Bregman perspective reframes true scores, error, and reliability as geometric constructs spanning both classical and modern psychometric models.

IRT as Nonlinear Projection

The Bregman framework provides a natural bridge between test theory and modern nonlinear latent variable models, including IRT. CTT treats observed scores as linear functions of latent variables plus additive error—consistent with Euclidean geometry and the quadratic potential $\varphi (x)=\frac{1}{2}{x}^2$ . Nevertheless, such a linear framework cannot capture the nonlinear relationships between latent traits and observed responses that define IRT.

Bregman geometry generalizes this structure by allowing projection in a curved statistical space determined by a convex potential $\varphi$ . When $\varphi (x)=x\log x-x$ , the associated divergence

$$D_{\varphi }(y,\hat{y})=y\log (\frac{y}{\hat{y}})-y+\hat{y}$$

is the KL divergence, which underlies exponential-family models such as logistic and Poisson IRT. Here, the projection $E_{\varphi }[X\mid \mathcal{G}]$ minimizes expected information loss, not squared deviation, making latent-trait estimation in IRT a Bregman projection within an information-geometric space. The link functions in IRT arise naturally from this geometry. The canonical link of an exponential-family model corresponds to the gradient mapping $\nabla \varphi$ , defining dual coordinates where the projection is linear. In the Rasch model, for example, the logit link

$${\eta }_i=\log (\frac{p_i}{1-p_i})$$

is generated by the potential

$$\varphi (p)=p\log p+(1-p)\log (1-p).$$

In these dual coordinates, the model is linear: ${\eta }_i=\theta -b_i$ .

Mapping back to the probability scale reintroduces nonlinearity, illustrating that IRT models are linear in the dual space and nonlinear in the primal space—a defining feature of Bregman projections. This dual space structure explains why both maximum likelihood and expected a posteriori (EAP) estimation in IRT correspond to Bregman projections. The likelihood function measures fit as divergence between observed and model-implied probabilities, and its optimization minimizes expected divergence.

This same principle extends to multidimensional IRT, where the latent space may be curved but remains dually flat under information geometry. Viewed geometrically, IRT is a measurement model on a curved manifold induced by a convex potential, where estimation, prediction, and scoring correspond to projection operations that minimize expected divergence.

Each family of item response models corresponds to a distinct potential $\varphi$ , defining a specific notion of distance or discrepancy. The logistic model corresponds to the negative binary entropy potential; Poisson or count-based models arise from the potential generating KL divergence; and graded response models derive from alternative convex potentials capturing the data’s distributional shape. In every case, estimation and scoring locate the point in the latent space that minimizes expected divergence from the observed data.

Reliability as Projection Efficiency

Within this framework, reliability generalizes as projection efficiency under divergence geometry. In CTT, reliability represents the proportion of variance explained by the projection $T=\mathbb{E}(X\mid \mathcal{G})$ . In the Bregman setting, the generalized projection

$$T_{\varphi }=E_{\varphi }[X\mid \mathcal{G}]=\arg \underset{Y\in L^1(\mathcal{G})}{\min }\mathbb{E}[D_{\varphi }(Y,X)]$$

leads to a reliability measure

$$\mathrm{Re}{\mathrm{l}}_{\varphi }(X)=1-\frac{\mathbb{E}[D_{\varphi }(X,T_{\varphi })]}{\mathbb{E}[D_{\varphi }(X,\mathbb{E}[X])]},$$

which reduces to the variance ratio when $\varphi (x)=\frac{1}{2}{x}^2$ .

For other potentials, $\mathrm{Re}{\mathrm{l}}_{\varphi }(X)$ captures the efficiency of projection under non-quadratic loss structures—linking reliability directly to information-theoretic precision.

In IRT, this interpretation connects reliability to the Fisher information function, which quantifies local curvature in the model’s information geometry. Greater curvature corresponds to higher sensitivity to changes in the latent coordinate $\theta$ and, therefore, greater measurement precision. The marginal reliability commonly reported in IRT—defined as the ratio of the variance of posterior mean scores to the total variance of the latent distribution—thus represents the average projection efficiency across the population of latent traits.

Both variance-based reliability in CTT and information-based reliability in IRT measure the same geometric quantity: alignment between observed and latent representations. In Euclidean geometry, this alignment is measured by squared deviation; in information geometry, by divergence. From this unified perspective, reliability is not a model-specific coefficient but a geometric property of measurement—the degree to which observed data lie near their theoretical projections in the geometry defined by the model’s convex potential. This view provides a coherent mathematical foundation linking classical and modern psychometric theory, situating both within a continuous family of linear and nonlinear measurement geometries.

Ordinal Reliability and Underlying Variable Models in the Bregman Framework

The concept of ordinal reliability, developed from underlying variable models and polychoric correlations (Zumbo et al., 2007), aligns naturally with the geometry of Bregman projections.

In these models, each observed ordinal score $X_i$ arises from a latent continuous variable $Z_i$ through thresholding:

$$X_i=k\mathrm{if}{\tau }_{k-1}<Z_i\le {\tau }_k,$$

where ${\tau }_k$ denotes ordered cut points on the latent continuum. The latent vector $Z$ is typically multivariate normal, and its correlations form the polychoric correlation matrix that defines the covariance structure of the latent Euclidean space. Within this space, reliability is the efficiency of projection—the same geometric principle that underlies CTT.

In the Bregman framework, this latent variable formulation embeds the ordinal response space within a curved manifold induced by a convex potential $\varphi$ . The ordinal scores themselves do not occupy a linear metric space, but their latent variables do. The mapping from $Z_i$ to $X_i$ is a nonlinear transformation determined by the cumulative distribution function (CDF) of $Z_i$ , corresponding to the gradient mapping $\nabla \varphi$ that defines the dual coordinate system in Bregman geometry. Thus, the latent variable represents the linear (dual) coordinates, and the observed ordinal score represents its nonlinear (primal) counterpart.

From this perspective, ordinal reliability reflects projection efficiency in the latent Euclidean space that best approximates the observed ordinal responses under the divergence induced by $\varphi$ .

The polychoric correlation matrix estimates this latent covariance structure by identifying the projection of the ordinal data that minimizes expected divergence from a multivariate normal model. Computing ordinal reliability through polychoric correlations is therefore equivalent to performing a Bregman projection from discrete ordinal data onto a continuous latent Gaussian manifold. The resulting coefficient quantifies how efficiently the observed ordinal indicators align with their latent projections—analogous to variance-based reliability in $L^2$ .

This geometric interpretation also provides a foundation for Zumbo’s (in press-b) treatment of polychoric and copula-based estimators of composite reliability. These estimators reconstruct an underlying continuous geometry consistent with a particular potential—quadratic for Gaussian copulas or more general convex functions for other dependence structures. Each potential defines a specific notion of “distance” or divergence appropriate to the measurement scale and data distribution.

Hence, ordinal reliability is not merely an empirical correction for discretization but a geometric projection problem. The thresholds defining ordinal categories act as nonlinear coordinate boundaries, and polychoric correlations reconstruct the latent inner products that determine projection efficiency. The observed ordinal space is curved, but the latent space remains linear, and reliability quantifies how well projections from one to the other preserve alignment.

This interpretation parallels the treatment of categorical responses in IRT. Both frameworks describe measurement as a projection from a discrete observed space onto a continuous latent manifold defined by a convex potential. In IRT, this projection occurs through logit or probit link functions, representing the gradient transformation $\nabla \varphi$ associated with an entropy-based potential such as

$$\varphi (p)=p\log p+(1-p)\log (1-p).$$

In the ordinal model, the thresholds partition the latent distribution analogously, producing observed categories that are nonlinear manifestations of an underlying linear geometry.

In both cases, estimation seeks the Bregman projection that minimizes expected divergence between observed responses and the latent model. Thus, ordinal reliability and IRT reliability express the same geometric principle.

Each quantifies how efficiently observed categorical responses align with their latent projections within the model’s information geometry. The polychoric approach reconstructs the latent covariance structure, whereas IRT estimation identifies the parameters defining the latent traits and response functions.

In both frameworks, reliability measures the degree to which nonlinear mappings from latent to observed space preserve geometric alignment—whether expressed as variance explained in $L^2$ or divergence minimized under $D_{\varphi }$ . This unified view underscores the generality of the Bregman perspective: whether data are continuous, ordinal, or dichotomous, measurement remains a problem of projection efficiency in a space shaped by the chosen convex potential.

As an overview, ordinal reliability, developed through underlying variable models and polychoric correlations, aligns with Bregman geometry. Ordinal scores map nonlinearly from latent continuous variables through threshold functions, analogous to transformations between primal and dual spaces. Polychoric correlations estimate the covariance structure of the latent space by identifying projections minimizing expected divergence. Thus, ordinal reliability represents projection efficiency in a latent convex geometry, linking seamlessly with the IRT formulation.

Concluding Remarks

I advance a unified geometric framework for psychometric theory. It connects CTT, operator-theoretic test theory, IRT, regression, and reliability through the mathematical construct of Bregman projections. The paper positions measurement as a projection problem within convex and information-geometric spaces, drawing on functional analysis, convex analysis, and information geometry to reinterpret long-standing psychometric concepts.

The main contribution lies not in new data or simulation results but in theoretical synthesis—clarifying that expectation, regression, and reliability are unified by a shared geometric principle. The work extends the author’s prior operator-theoretic research (Zumbo, in press-a) and aligns with ongoing theoretical modernization in measurement science.

The Bregman framework provides a geometric foundation for understanding expectation, regression, and reliability within a single theoretical structure. By replacing the quadratic potential that defines Euclidean geometry with a general convex potential, this approach encompasses a wide class of statistical and psychometric models. Conditional expectation, least-squares regression, and classical reliability emerge as special cases of Bregman projections under the quadratic potential $\varphi (x)=\frac{1}{2}{x}^2$ , while other potentials yield models grounded in exponential, multiplicative, or logistic geometries.

This unifying view clarifies that many methods in psychometrics—such as regression, factor analysis, item response theory, and composite reliability estimation—share a common mathematical foundation: projection under a divergence defined by a convex potential. Reliability, for example, quantifies the efficiency of this projection and generalizes naturally across linear and nonlinear models. Regression likewise represents the Bregman projection that minimizes expected divergence between observed and predicted values.

By situating CTT within the broader class of Bregman geometries, reliability emerges not as a static parameter tied to linear models but as a projection efficiency that depends on the chosen geometry of measurement and strengthens conceptual links between classical and modern measurement theories.

Bregman projections thus offer a unifying principle that connects the statistical and geometric foundations of measurement. They demonstrate that reliability, estimation, and model fit can each be understood as instances of a broader projection framework, governed by the choice of divergence and its associated geometry. This view situates psychometric theory within a coherent mathematical structure that generalizes beyond linear models.

It unifies the Hilbert-space representations of test theory with the information-geometric frameworks that underpin modern statistical learning. In this perspective, measurement error is understood as the residual divergence of a projection, and reliability expresses the efficiency with which observed data align with latent structures. Bregman geometry thereby provides a rigorous mathematical formalization of long-standing psychometric concepts while preserving their substantive interpretability.

By extending measurement theory into nonlinear and information-geometric domains, this approach suggests that psychometric quantities are not fixed properties of models but context-dependent expressions of geometric relationships. Such a reconceptualization opens a path toward integrating modern machine learning, information theory, and CTT under a common projection-based framework.

Appendices A and B summarize the mathematical details supporting the operator-theoretic and geometric results. Appendix A outlines the formal properties of the generalized conditional expectation operator $E_{\varphi }[X\mid \mathcal{G}]$ , including idempotence, minimality, and generalized orthogonality under the Bregman divergence $D_{\varphi }$ . Appendix B establishes that the proposed reliability measure $\mathrm{Re}{\mathrm{l}}_{\varphi }(X)$ preserves key theoretical properties of its classical counterpart—boundedness, monotonicity under finer conditioning, and reduction to the variance-based ratio when $\varphi (x)=\frac{1}{2}{x}^2$ . Together, these results confirm that the Bregman formulation retains the foundational structure of classical test theory while extending it coherently to nonlinear and information-geometric settings.

From a practical standpoint, the Bregman perspective offers new pathways for modeling data in contexts where nonlinear relationships arise naturally from ordinal response formats, bounded rating scales, or skewed score distributions. It supports the use of polychoric, copula-based, and entropy-based estimators as natural extensions of Euclidean reliability measures (Zumbo, in press-b). More broadly, this framework invites a reexamination of measurement models through the lens of geometry, revealing reliability, regression, and estimation as manifestations of the same underlying principle: projection in spaces shaped by convex potentials.