Collaborate Research: Construct a General Hilbert Space Multi-dimensional Model

该研究项目将开发和测试一种基于量子概率论的新的测量模型，称为希尔伯特空间多维模型。随着现代数据收集方法的显著进步，从概念上相互关联的各种来源和上下文生成复杂而大量的数据集。这有望为复杂的社会和行为现象提供更好的理解，但它也为整合和解释来自多个来源的数据提出了重大挑战。一般希尔伯特空间多维模型将提高对复杂社会和行为现象的理解，从违反理性决策理论到社会调查数据的整合和解释。该项目是一个更大的研究计划的一部分，该计划旨在从量子而不是经典概率原理为社会和行为科学建立概率和动态系统。该项目将从公共存储库开发和传播用于在MATLAB、R和Python中应用和估计一般Hilbert空间多维模型的自包含软件包。研究人员将开发和测试一般希尔伯特空间多维模型，包括开发该模型的数学理论以及应用该模型的相关统计和计算工具。他们将通过大范围的实验严格检验这个模型。当从不同的上下文或条件收集大型数据集时，通常可以通过列联表对它们进行汇总。然而，出现了一个关键问题，即如何将这些表集成和合成为压缩的、连贯的和可解释的表示。一个常见的解决方案是尝试构造一个联合概率分布来重现表中观察到的频率数据。贝叶斯因果网络通常通过施加条件独立性假设来减少估计参数的数量。然而，在许多情况下，不存在能够再现所观察到的表的这种联合分布。一般希尔伯特空间多维模型通过构造位于低维希尔伯特空间内的单一有限状态向量，并形成一组表示测量值的非交换测量算子，为复杂和海量数据所面临的问题提供了一个有希望的解决方案。通过这种方式，即使在不存在标准联合分布的情况下，模型也可以对构成数据表复杂集合的测量变量产生压缩的、连贯的和可解释的表示。

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.