Hilbert Space Multi-dimensional Modeling

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.