Abstract
This chapter provides a brief introduction to procedures for estimating Hilbert space multi-dimensional (HSM) models from data. These models, which are built from quantum probability theory, are used to provide a simple and coherent account of a collection of contingency tables. The collection of tables are obtained by measurement of different overlapping subsets of variables. HSM models provide a representation of the collection of the tables in a low dimensional vector space, even when no single joint probability distribution across the observed variables can reproduce the tables. The parameter estimates from HSM models provide simple and informative interpretation of the initial tendencies and the inter-relations among the variables.
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Notes
- 1.
- 2.
Technically, a Hilbert space is a complex valued inner product vector space that is complete. Our vectors spaces are finite, and so they are always complete.
- 3.
A more general approach uses what is called a density operator rather than a pure state vector, but to keep ideas simple, we use the latter.
- 4.
∪ is the union of subsets A, B.
- 5.
∨ is the span of subspaces A, B.
References
Abramsky, S. (2013). Relational databases and Bells theorem. In V. Tannen, L. Wong, L. Libkin, W. Fan, T. Wang-Chiew, & M. Fourman (Eds.), In search of elegance in the theory and practice of computation (pp. 13–35). Berlin: Springer.
Aerts, D., & Aerts, S. (1995). Applications of quantum statistics in psychological studies of decision processes. Foundations of Science, 1(1), 85–97.
Agresti, A., & Katera, M. (2011). Categorical data analysis. Berlin: Springer.
Atmanspacher, H., Römer, H., & Walach, H. (2002). Weak quantum theory: Complementarity and entanglement in physics and beyond. Foundations of Physics, 32(3), 379–406.
Bordley, R. F. (1998). Quantum mechanical and human violations of compound probability principles: Toward a generalized Heisenberg uncertainty principle. Operations Research, 46(6), 923–926.
Bruza, P. D., Kitto, K., Ramm, B. J., & Sitbon, L. (2015). A probabilistic framework for analysing the compositionality of conceptual combinations. Journal of Mathematical Psychology, 67, 26–38.
Busemeyer, J. R., & Bruza, P. D. (2012). Quantum models of cognition and decision. Cambridge: Cambridge University Press.
Busemeyer, J. R., & Wang, Z. (2018). Data fusion using Hilbert space multi-dimensional models. Theoretical Computer Science, 752, 41–55.
Busemeyer, J. R. & Wang, Z. (2018). Hilbert space multidimensional theory. Psychological Review, 125(4), 572–591.
Darwiche, A. (2009). Modeling and reasoning with Bayesian networks. New York, NY: Cambridge University Press.
Dzhafarov, E., & Kujala, J. V. (2012). Selectivity in probabilistic causality: Where psychology runs into quantum physics. Journal of Mathematical Psychology, 56, 54–63.
Gleason, A. M. (1957). Measures on the closed subspaces of a Hilbert space. Journal of Mathematical Mechanics, 6, 885–893.
Gudder, S. P. (1988). Quantum probability. Boston, MA: Academic Press.
Khrennikov, A. (1999). Classical and quantum mechanics on information spaces with applications to cognitive, psychological, social, and anomalous phenomena. Foundations of Physics, 29(7), 1065–1098.
Khrennikov, A. Y. (2010). Ubiquitous quantum structure: From psychology to finance. Berlin: Springer.
Melucci, M. (2015). Introduction to information retrieval and quantum mechanics. Berlin: Springer.
Pothos, E. M., Busemeyer, J. R., & Trueblood, J. S. (2013). A quantum geometric model of similarity. Psychological Review, 120(3), 679–696
van Rijsbergen, C. J. (2004). The geometry of information retrieval. Cambridge: Cambridge University Press.
Wang, Z., & Busemeyer, J. R. (2016). Comparing quantum versus Markov random walk models of judgments measured by rating scales. Philosophical Transactions of the Royal Society, A, 374, 20150098.
Acknowledgements
This research was based upon the work supported by NSF SES-1560501 and NSF SES-1560554.
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Busemeyer, J., Wang, Z.J. (2019). Introduction to Hilbert Space Multi-Dimensional Modeling. In: Aerts, D., Khrennikov, A., Melucci, M., Toni, B. (eds) Quantum-Like Models for Information Retrieval and Decision-Making. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-030-25913-6_3
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