Reliability as Projection in Operator-Theoretic Test Theory: Conditional Expectation, Hilbert Space Geometry, and Implications for Psychometric Practice.

We review the construction of monsters in classical general relativity. Monsters have finite ADM mass and surface area, but potentially unbounded entropy. From the curved space perspective, they are objects with large proper volume that can be glued on to an asymptotically flat space. At no point is the curvature or energy density required to be large in Planck units, and quantum gravitational effects are, in the conventional effective field theory framework, small everywhere. Since they can have more entropy than a black hole of equal mass, monsters are problematic for certain interpretations of black hole entropy and the AdS/CFT duality. In the second part of the paper we review recent developments in the foundations of statistical mechanics which make use of properties of high-dimensional (Hilbert) spaces. These results primarily depend on kinematics — essentially, the geometry of Hilbert space — and are relatively insensitive to dynamics. We discuss how this approach might be adopted as a basis for the statistical mechanics of gravity. Interestingly, monsters and other highly entropic configurations play an important role.

A generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product. Complementation is a construction in the geometry of Hilbert space which was applied to the invariant subspace theory of contractive transformations in Hilbert space by James Rovnyak and the author [6]. The concept was later formalized by the author [3]. Continuous and contractive transformations in Krein spaces appear in the estimation theory of Riemann mapping functions [4]. It is therefore of interest to know whether a generalization of complementation theory applies in Krein spaces. Such a generalization is now obtained. The results are also of interest in the invariant subspace theory of continuous and contractive transformations in Krein spaces [5]. The vector spaces considered are taken over the complex numbers. A scalar product for a vector space )1 is a complex-valued function (a, b) of a and b in )1 which is linear, symmetric, and nondegenerate. Linearity means that the identity (aa + 3b, c) = a(a, c) + 3(b, c) holds for all elements a, b, and c of )1 when a and , are complex numbers. Symmetry means that the identity (b, a) = (a, b) holds for all elements a and b of )1. Nondegeneracy means that an element a of )1 is zero if the scalar product (a, b) is zero for every element b of )1. Every element b of )1 determines a linear functional bon )1 which is defined by b-a = (a, b) for every element a of )1. The weak topology of )1 is the weakest topology with respect to which bis a continuous linear functional on )1 for every element b of )1. The weak topology of )1 is a locally convex topology having the property that every continuous linear functional on )1 is of the form bfor an element b of )1. The element b is then unique. The antispace of a vector space with scalar product is the same vector space considered with the negative of the given scalar product. A fundamental example of a vector space with scalar product is a Hilbert space. A Krein space is a vector space with scalar product which is the orthogonal sum of a Hilbert space and the antispace of a Hilbert space. Received by the editors October 10, 1986 and, in revised form, February 23, 1987. The results of the paper were presented to the Department of Mathematics, Indiana and Purdue University in Indianapolis, on March 27, 1987, as the Ernest J. Johnson Colloquium. 1980 Mathematics Subject Classification (1985 Revision). Primary 46D05. Research supported by the National Science Foundation. ?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

A branch of probability theory that has been studied extensively in recent years, the theory of conditional expectation, provides just the concepts needed for mathematical derivation of the main results of the classical test theory with minimal assumptions and greatest economy in the proofs. The collection of all random variables with finite variance defined on a given probability space is a Hilbert space; the function that assigns to each random variable its conditional expectation is a linear operator; and the properties of the conditional expectation needed to derive the usual test-theory formulas are general properties of linear operators in Hilbert space. Accordingly, each of the test-theory formulas has a simple geometric interpretation that holds in all Hilbert spaces.

In this paper, we study the geometry of quadratic covariance bounds on the estimation error covariance, in a properly defined Hilbert space of random variables. We show that a lower bound on the error covariance may be represented by the Grammian of the error score after projection onto the subspace spanned by the measurement scores. The Grammian is defined with respect to inner products in a Hilbert space of second order random variables. This geometric result holds for a large class of quadratic covariance bounds including the Barankin, Cramer-Rao, and Bhattacharyya bounds, where each bound is characterized by its corresponding measurement scores. When parameters consist of essential parameters and nuisance parameters, the Cramer-Rao covariance bound is the inverse of the Grammian of essential scores after projection onto the subspace orthogonal to the subspace spanned by the nuisance scores. In two examples, we show that for complex multivariate normal measurements with parameterized mean or covariance, there exist well-known Euclidean space geometries for the general Hilbert space geometry derived in this paper.

Human feedback is often used, either directly or indirectly, as input to algorithmic decision making. However, humans are biased: if the algorithm that takes as input the human feedback does not control for potential biases, this might result in biased algorithmic decision making, which can have a tangible impact on people’s lives. In this paper, we study how to detect and correct for evaluators’ bias in the task of ranking people (or items) from pairwise comparisons. Specifically, we assume we are given pairwise comparisons of the items to be ranked produced by a set of evaluators. While the pairwise assessments of the evaluators should reflect to a certain extent the latent (unobservable) true quality scores of the items, they might be affected by each evaluator’s own bias against, or in favor, of some groups of items. By detecting and amending evaluators’ biases, we aim to produce a ranking of the items that is, as much as possible, in accordance with the ranking one would produce by having access to the latent quality scores. Our proposal is a novel method that extends the classic Bradley-Terry model by having a bias parameter for each evaluator which distorts the true quality score of each item, depending on the group the item belongs to. Thanks to the simplicity of the model, we are able to write explicitly its log-likelihood w.r.t. the parameters (i.e., items’ latent scores and evaluators’ bias) and optimize by means of the alternating approach. Our experiments on synthetic and real-world data confirm that our method is able to reconstruct the bias of each single evaluator extremely well and thus to outperform several non-trivial competitors in the task of producing a ranking which is as much as possible close to the unbiased ranking.

Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on a Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, \ldots, T_n)$ turns $\mathcal{H}$ into a module over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. The above module is usually called the Hilbert module over $\mathbb{C}[z_1, \ldots, z_n]$. Hilbert modules over $\mathbb{C}[z_1, \ldots, z_n]$ (or natural function algebras) were first introduced by R. G. Douglas and C. Foias in 1976. The two main driving forces were the algebraic and complex geometric views to multivariable operator theory. This article gives an introduction of Hilbert modules over function algebras and surveys some recent developments. Here the theory of Hilbert modules is presented as combination of commutative algebra, complex geometry and the geometry of Hilbert spaces and its applications to the theory of $n$-tuples ($n \geq 1$) of commuting operators. The topics which are studied include: model theory from Hilbert module point of view, Hilbert modules of holomorphic functions, module tensor products, localizations, dilations, submodules and quotient modules, free resolutions, curvature and Fredholm Hilbert modules. More developments in the study of Hilbert module approach to operator theory can be found in a companion paper, Applications of Hilbert Module Approach to Multivariable Operator Theory.

In order to estimate the causal effects of one or more exposures or treatments on an outcome of interest, one has to account for the effect of "confounding factors" which both covary with the exposures or treatments and are independent predictors of the outcome. In this paper we present regression methods which, in contrast to standard methods, adjust for the confounding effect of multiple continuous or discrete covariates by modelling the conditional expectation of the exposures or treatments given the confounders. In the special case of a univariate dichotomous exposure or treatment, this conditional expectation is identical to what Rosenbaum and Rubin have called the propensity score. They have also proposed methods to estimate causal effects by modelling the propensity score. Our methods generalize those of Rosenbaum and Rubin in several ways. First, our approach straightforwardly allows for multivariate exposures or treatments, each of which may be continuous, ordinal, or discrete. Second, even in the case of a single dichotomous exposure, our approach does not require subclassification or matching on the propensity score so that the potential for "residual confounding," i.e., bias, due to incomplete matching is avoided. Third, our approach allows a rather general formalization of the idea that it is better to use the "estimated propensity score" than the true propensity score even when the true score is known. The additional power of our approach derives from the fact that we assume the causal effects of the exposures or treatments can be described by the parametric component of a semiparametric regression model. To illustrate our methods, we reanalyze the effect of current cigarette smoking on the level of forced expiratory volume in one second in a cohort of 2,713 adult white males. We compare the results with those obtained using standard methods.

Reliability: What type, please!

We adopt and expand McDonald's (2011) regression framework for measurement precision, integrating two key perspectives: (a) reliability of observed scores and (b) optimal prediction of latent scores. Reliability arises from a measurement decomposition of an observed score into its true score and measurement error. In contrast, proportional reduction in mean squared error (PRMSE) arises from a prediction decomposition of a latent score into its optimal predictor (the observed expected a posteriori [EAP] score) and prediction error. Reliability is the coefficient of determination obtained by two isomorphic regressions: regressing the observed score on its true score or on all the latent variables. Similarly, PRMSE is the coefficient of determination obtained from two isomorphic regressions: regressing the latent score on its observed EAP score or all the manifest variables. A key implication of this regression framework is that both reliability and PRMSE can be estimated using a Monte Carlo (MC) method, which is particularly useful when no analytic formula is available or when the analytic calculation is involved. We illustrate these concepts with a factor analysis model and a two-parameter logistic model, in which we compute reliability coefficients for different observed scores and PRMSE for different latent scores. Additionally, we provide a numerical example demonstrating how the MC method can be used to estimate reliability and PRMSE within a two-dimensional item response tree model. (PsycInfo Database Record (c) 2025 APA, all rights reserved).

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. One is a vector conditional expectation defined by a linear map of anO∗-algebra into the Hilbert space on which theO∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear mapℰof anO∗-algebraℳonto a givenO∗-subalgebra𝒩ofℳ. Here the domainD(ℰ)ofℰdoes not equal toℳin general, and so such a conditional expectation is called unbounded.

Two concepts of “true score” in test theory are examined. Under one concept, the true score is identified with the expected value of the observed score, and it follows that reliability is the ratio of true variance to observed variance. Under the other concept, the true score is a constant which is not necessarily equal to the expected value of the observed score, and it follows that reliability is not necessarily equal to the ratio of true variance to observed variance. Axioms are presented which encompass both points of view, and explicit formulas relating the two kinds of true scores are derived by representing all scores and components of scores as random variables with the same associated probability space.

On Some Limit Theorems Similar to the Arc-Sin Law

The concept of a superposition is a revolutionary novelty introduced by Quantum Mechanics. If a system may be in any one of two pure states x and y, we must consider that it may also be in any one of many superpositions of x and y. An in-depth analysis of superpositions is proposed, in which states are represented by one-dimensional subspaces, not by unit vectors as in Dirac’s notation. Superpositions must be considered when one cannot distinguish between possible paths, i.e., histories, leading to the current state of the system. In such a case the resulting state is some compound of the states that result from each of the possible paths. States can be compounded, i.e., superposed in such a way only if they are not orthogonal. Since different classical states are orthogonal, the claim implies no non-trivial superpositions can be observed in classical systems. The parameter that defines such compounds is a proportion defining the mix of the different states entering the compound. Two quantities, p and θ, both geometrical in nature, relate one-dimensional subspaces in complex Hilbert spaces: the first one is a measure of proximity relating two rays, the second one is an angle relating three rays. The properties of superpositions with respect to those two quantities are studied. The algebraic properties of the operation of superposition are very different from those that govern linear combination of vectors.

This chapter provides the setting and framework for the rest of the book. It is divided into three parts. The first part concerns itself with the geometry of Hilbert space. The second part gives the basic results in operator theory needed throughout this book, and the third part provides a short introduction to the theory of Banach algebras.

We have seen that the equivalence of state vectors which differ by a phase, along with the scalar product, define the geometry of Hilbert space (i.e. the fiber bundle and connection). The geometry is non-trivial. It induces a U(1) holonomy in a normalized state vector which undergoes cyclic evolution. This induced phase is called the geometric phase. It depends only on the path in the space of physical states, not on the Hamiltonian which generates this path.