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理论的新的测量模型，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，从概念上联系在一起的各种来源和背景中产生了复杂和海量的数据集。这有望更好地理解复杂的社会和行为现象，但它也给来自多个来源的数据的整合和解释带来了重大挑战。通用的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。这个项目是一个更大的研究项目的一部分，该项目旨在根据量子而不是经典的概率原理为社会和行为科学构建概率和动态系统。该项目将在公共资源库中开发和分发用于应用和估计通用希尔伯特空间多维模型的自备软件包。研究人员将开发和测试通用希尔伯特空间多维模型，包括该模型的数学理论的发展以及应用该模型的相关统计和计算工具。他们将使用大范围的实验来严格测试该模型。当从不同的环境或条件收集大型数据集时，通常可以通过联想表对其进行汇总。然而，关于如何将这些表格集成和合成成压缩的、连贯的和可解释的表示，出现了一个关键问题。一种常见的解决方案是尝试构建联合概率分布，以重现表格中观察到的频率数据。然后，贝叶斯因果网络经常被用来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，不存在能够再现观察到的表格的这种联合分布。一般的希尔伯特空间多维模型通过构造一个位于低维希尔伯特空间的有限状态向量，并通过形成一组表示测量的非交换测量算子，为解决复杂和海量数据所面临的问题提供了一种有前途的解决方案。通过这种方式，即使在不存在标准联合分布的情况下，该模型也产生了对测量变量的压缩、连贯和可解释的表示，这些变量形成了复杂的数据表集合。

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.