Hilbert Space Multi-dimensional Modeling

TL;DR

This paper introduces Hilbert space multi-dimensional (HSM) models based on quantum probability theory, enabling the analysis of multiple contingency tables from different contexts, even without a joint distribution, with broad applications in social sciences.

Contribution

The paper develops general procedures for constructing, estimating, and testing HSM models, demonstrating their viability and advantages over Bayes net models in social science data analysis.

Findings

HSM models effectively represent collections of contingency tables in low-dimensional space.

HSM models outperform Bayes net models in fit for large social science datasets.

HSM models provide interpretable parameters for complex multi-table data.

Abstract

This article presents general procedures for constructing, estimating, and testing Hilbert space multi-dimensional (HSM) models, which are based on quantum probability theory. HSM models can be applied to collections of K different contingency tables obtained from a set of p variables that are measured under different contexts. A context is defined by the measurement of a subset of the p variables that are used to form a table. HSM models provide a representation of the collection of K tables in a low dimensional vector space, even when no single joint probability distribution across the p variables exists. HSM models produce parameter estimates that provide a simple and informative interpretation of the complex collection of tables. Comparisons of HSM model fits with Bayes net model fits are reported for a new large experiment, demonstrating the viability of this new model. We conclude that the model is broadly applicable to social and behavioral science data sets.