This research project will develop and test a new measurement model based on quantum probability theory called the Hilbert space multi-dimensional model. With the striking advancement of modern data-collection methods, complex and massive data sets are generated from various sources and contexts that are conceptually connected. This promises to provide a better understanding of complex social and behavioral phenomena, but it also presents significant challenges for the integration and interpretation of data from multiple sources. The general Hilbert space multi-dimensional model will improve understanding of complex social and behavioral phenomena ranging from violations of rational decision theory to social survey data integration and interpretation. This project is part of a larger research program to build probabilistic and dynamic systems for social and behavioral sciences from quantum rather than classical probability principles. The project will develop and disseminate from public repositories self-contained software packages for applying and estimating the general Hilbert space multi-dimensional model in MATLAB, R, and Python.
The investigators will develop and test the general Hilbert space multi-dimensional model, including the development of the mathematical theory of the model and related statistical and computational tools for applying the model. They will rigorously test the model using a large range of experiments. When large data sets are collected from different contexts or conditions, often they can be summarized by contingency tables. A critical problem arises, however, regarding how to integrate and synthesize these tables into a compressed, coherent, and interpretable representation. A common solution is to try to construct a joint probability distribution to reproduce the frequency data observed in the tables. Bayesian causal networks then are often used to reduce the number of estimated parameters by imposing conditional independence assumptions. In many cases, however, no such joint distribution exists that can reproduce the observed tables. The general Hilbert space multi-dimensional model provides a promising solution to the problems faced by complex and massive data by constructing a single finite state vector that lies within a low dimensional Hilbert space and by forming a set of non-commuting measurement operators that represent the measurements. In this way, the model produces a compressed, coherent, and interpretable representation of the measured variables that form the complex collection of data tables even when no standard joint distribution exists.