Probability Spaces and the Theory of Error of Measurement

1971 ◽  
Vol 28 (1) ◽  
pp. 291-301 ◽  
Author(s):  
Donald W. Zimmerman
Keyword(s):  
Probability Model ◽  
Random Variables ◽  
Measurement Procedure ◽  
Psychological Tests ◽  
Random Variability ◽  
Probability Spaces ◽  
True Values ◽  
Suitable Product ◽  
Error Of Measurement

A model of variability in measurement, which is sufficiently general for a variety of applications and which includes the main content of traditional theories of error of measurement and psychological tests, can be derived from the axioms of probability, without introducing “true values” and “errors.” Beginning with probability spaces (Ω, P1) and (φ, P2), the set Ω representing the outcomes of a measurement procedure and the set * representing individuals or experimental objects, it is possible to construct suitable product probability spaces and collections of random variables which can yield all results needed to describe random variability and reliability. This paper attempts to fill gaps in the mathematical derivations in many classical theories and at the same time to overcome limitations in the language of “true values” and “errors” by presenting explicitly the essential constructions required for a general probability model.

Error and Reliability in Stochastic Processes and Psychological Measurement

1972 ◽  
Vol 31 (1) ◽  
pp. 131-140 ◽  
Author(s):  
Donald W. Zimmerman
Keyword(s):  
Stochastic Processes ◽  
Random Error ◽  
Random Sampling ◽  
Probability Model ◽  
Random Variables ◽  
Formal Structure ◽  
Intermediate Structure ◽  
Psychological Measurement ◽  
Joint Distributions ◽  
True Values

The concepts of random error and reliability of measurements that are familiar in traditional theories based on the notions of “true values” and “errors” can be represented by a probability model having a simpler formal structure and fewer special assumptions about random sampling and independence of measurements. In this model formulas that relate observable events are derived from probability axioms and from primitive terms that refer to observable events, without an intermediate structure containing variances and correlations of “true” and “error” components of scores. While more economical in language and formalism, the model at the same time is more general than classical theories and applies to stochastic processes in which joint distributions of many dependent random variables are of interest. In addition, it clarifies some long-standing problems concerning “experimental independence” of measurements and the relation of sampling of individuals to sampling of measurements.

A Simplified Probability Model of Error of Measurement

1969 ◽  
Vol 25 (1) ◽  
pp. 175-186 ◽  
Author(s):  
Donald W. Zimmerman
Keyword(s):  
Present Model ◽  
Probability Model ◽  
Classical Test Theory ◽  
Random Variable ◽  
Psychological Theory ◽  
Testing Procedure ◽  
Test Theory ◽  
Observed Score ◽  
True Values ◽  
Error Of Measurement

A model of variability in measurement which does not employ the concepts of “true score” and “error score” is presented. Reference to an observed score random variable, X, together with the usual axioms of probability, is shown to be a satisfactory basis for derivation of results of the classical test theory which relate observable quantities. In addition, reliability formulas such as the KR 20 and KR 21 are obtained by construction of the observed score random variable over a sample space of outcomes of a testing procedure and assignment of probabilities to outcomes. The approach is consistent with trends in psychological theory toward objectively defined constructs and avoids redundancy in derivations, as well as connotations which arise from reference to “true values” and “errors.” The present model is shown to be consistent with a relativistic, as opposed to an absolutistic, conception of measurement.

A normal form theorem for Lω1p, with applications

1982 ◽  
Vol 47 (3) ◽  
pp. 605-624 ◽  
Author(s):  
Douglas N. Hoover
Keyword(s):  
Normal Form ◽  
Probability Model ◽  
Random Variables ◽  
Interpolation Theorem ◽  
Complete Theory ◽  
De Finetti’S Theorem ◽  
Form Theorem ◽  
Normal Form Theorem ◽  
De Finetti's Theorem

AbstractWe show that every formula of Lω1P is equivalent to one which is a propositional combination of formulas with only one quantifier. It follows that the complete theory of a probability model is determined by the distribution of a family of random variables induced by the model. We characterize the class of distribution which can arise in such a way. We use these results together with a form of de Finetti’s theorem to prove an almost sure interpolation theorem for Lω1P.

Probability Spaces and Random Variables

1989 ◽  
pp. 70-95
Author(s):  
J. Stoyanov ◽  
I. Mirazchiiski ◽  
Z. Ignatov ◽  
M. Tanushev
Keyword(s):  
Random Variables ◽  
Probability Spaces

scholarly journals Classical (Local and Contextual) Probability Model for Bohm–Bell Type Experiments: No-Signaling as Independence of Random Variables

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 157 ◽  
Author(s):  
Andrei Khrennikov ◽  
Alexander Alodjants
Keyword(s):  
Conditional Probability ◽  
Probability Model ◽  
Hidden Variables ◽  
Random Variables ◽  
Experimental Scheme ◽  
Model Following ◽  
No Signaling ◽  
Independence Of Random Variables ◽  
Main Message ◽  
Selection Of

We start with a review on classical probability representations of quantum states and observables. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The main part of the paper is devoted to the conditional probability model with conditioning on the selection of the pairs of experimental settings. From the viewpoint of quantum foundations, this is a local contextual hidden-variables model. Following the recent works of Dzhafarov and collaborators, we apply our conditional probability approach to characterize (no-)signaling. Consideration of the Bohm–Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science. The main message of this paper (rooted to Ballentine) is that quantum probabilities and more generally probabilities related to the Bohm–Bell type experiments (not only in physics, but also in psychology, sociology, game theory, economics, and finances) can be classically represented as conditional probabilities.

Asymmetry of Creative Activity: Product-Based Iterative Measurement Procedure

2010 ◽  
Vol 437 ◽  
pp. 520-524
Author(s):  
Lidia A. Mazhul ◽  
Vladimir M. Petrov ◽  
Stefania Mancone
Keyword(s):  
Measurement Procedure ◽  
Hemispheric Dominance ◽  
Relative Role ◽  
Creative Activity ◽  
Activity Product ◽  
Tuning Forks ◽  
Left And Right ◽  
Iterative Measurement ◽  
Error Of Measurement

To measure the relative role of left- or right-hemispheric processes involved both in creation of works of art and their perception, a procedure was derived based on calibrated expert estimations (of works or their authors). The resulting instrument for measurements consists of 7-10 scales and two sets of ‘contrastive objects’ (‘tuning forks,’ each containing 7-12 objects). The asymmetry is reflected by appropriate index which varies from –1 to +1 (‘pure’ right- and ‘pure’ left-hemispheric dominance, respectively). It occurs possible to eliminate subjectivity both of experts and researchers (compilers of the instrument). The method was used in evolutionary studies concerning 102 composers, 240 painters, etc.; the error of measurement was usually 3-5%. Evolutionary curves built on the basis of these indices, permitted to observe periodical switching between left and right styles in music, painting, theatre staging, and poetry.

scholarly journals A Unified Definition of Mutual Information with Applications in Machine Learning

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Guoping Zeng
Keyword(s):  
Machine Learning ◽  
Mutual Information ◽  
Joint Distribution ◽  
Credit Scoring ◽  
Random Variables ◽  
Joint Probability ◽  
Class 1 ◽  
Probability Spaces ◽  
Definition Of ◽  
Joint Probabilities

There are various definitions of mutual information. Essentially, these definitions can be divided into two classes: (1) definitions with random variables and (2) definitions with ensembles. However, there are some mathematical flaws in these definitions. For instance, Class 1 definitions either neglect the probability spaces or assume the two random variables have the same probability space. Class 2 definitions redefine marginal probabilities from the joint probabilities. In fact, the marginal probabilities are given from the ensembles and should not be redefined from the joint probabilities. Both Class 1 and Class 2 definitions assume a joint distribution exists. Yet, they all ignore an important fact that the joint or the joint probability measure is not unique. In this paper, we first present a new unified definition of mutual information to cover all the various definitions and to fix their mathematical flaws. Our idea is to define the joint distribution of two random variables by taking the marginal probabilities into consideration. Next, we establish some properties of the newly defined mutual information. We then propose a method to calculate mutual information in machine learning. Finally, we apply our newly defined mutual information to credit scoring.

scholarly journals Random Variables and Product of Probability Spaces

2013 ◽  
Vol 21 (1) ◽  
pp. 33-39
Author(s):  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Borel Sets ◽  
The Real ◽  
Probability Spaces ◽  
Countably Infinite

Summary We have been working on the formalization of the probability and the randomness. In [15] and [16], we formalized some theorems concerning the real-valued random variables and the product of two probability spaces. In this article, we present the generalized formalization of [15] and [16]. First, we formalize the random variables of arbitrary set and prove the equivalence between random variable on Σ, Borel sets and a real-valued random variable on Σ. Next, we formalize the product of countably infinite probability spaces.

Records, permutations and greatest convex minorants

1989 ◽  
Vol 106 (1) ◽  
pp. 169-177 ◽  
Author(s):  
Charles M. Goldie
Keyword(s):  
Random Walk ◽  
Random Variables ◽  
Random Permutations ◽  
Record Times ◽  
Distribution Free ◽  
Standard Representation ◽  
Bernoulli Random Variables ◽  
Convex Minorants ◽  
Probability Spaces

AbstractTheorems on random permutations are translated into distribution-free results about record times and greatest convex minorants, by defining them together on appropriate probability spaces. The Bernoulli random variables that appear in the standard representation of the number of sides of the greatest convex minorant of a random walk are identified.

scholarly journals Complete Convergence for END Random Variables under Sublinear Expectations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Qunying Wu
Keyword(s):  
Complete Convergence ◽  
Random Variables ◽  
Weighted Sums ◽  
Partial Sums ◽  
Convergence Theorems ◽  
Classical Probability ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Probability Spaces ◽  
End Random Variables

In this paper, the complete convergence theorems of partial sums and weighted sums for extended negatively dependent random variables in sublinear expectation spaces have been studied and established. Our results extend the corresponding results of classical probability spaces to the case of sublinear expectation spaces.

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