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arXiv:2310.12762

A new foundation of quantum decision theory

Inge Svein Helland

Published Oct 19, 2023• 0 citations
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Abstract

Quantum decision theory is introduced here, and new basis for this theory is proposed. It is first based upon a recent general arguments for the Hilbert space formalism in quantum theory, next on arguments for the Born rule, that is, the basis for calculating quantum probabilities. A fundamental notion behind the quantum theory foundation is that of theoretical variables, that are divided into accessible and inaccessible ones. This is here specialized to decision variables. A basic assumption is that there exist two complementary (non-equivalent, maximal accessible) theoretical variables in the given situation. Two basic assumptions behind the Born rule are 1) the likelihood principle, 2) the actor in question has motivations that can be modelled by a hypothetical perfectly rational higher being. It is argued that my way of thinking may elucidate a set of arguments for quantum-like models in cognition, as given by Busemeyer and Bruza (2012). Finally, a brief discussion of decision processes is given.

DOI
10.48550/arxiv.2310.12762

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Quantum decision theory is introduced here, and new basis for this theory is proposed. It is first based upon a recent general arguments for the Hilbert space formalism in quantum theory, next on arguments for the Born rule, that is, the basis for calculating quantum probabilities. A fundamental notion behind the quantum theory foundation is that of theoretical variables, that are divided into accessible and inaccessible ones. This is here specialized to decision variables. A basic assumption is that there exist two complementary (non-equivalent, maximal accessible) theoretical variables in the given situation. Two basic assumptions behind the Born rule are 1) the likelihood principle, 2) the actor in question has motivations that can be modelled by a hypothetical perfectly rational higher being. It is argued that my way of thinking may elucidate a set of arguments for quantum-like models in cognition, as given by Busemeyer and Bruza (2012). Finally, a brief discussion of decision processes is given.
Quantum decision theory is introduced here, and new basis for this theory is proposed. It is first based upon the author's general arguments for the Hilbert space formalism in quantum theory, next on arguments for the Born rule, that is the basis for calculating quantum probabilities. A basic notion behind the quantum theory foundation is that of theoretical variables, that are divided into accessible and inaccessible ones. This is here specialized to decision variables. It is assumed that each accessible variable can be seen as a function of a specific inaccessible variable. Another assumption is that there exist two maximal accessible decision processes in the given situation. Two basic assumptions behind the Born rule are 1) the likelihood principle, 2) the actor in question has motivations that can be modeled by a hypothetical perfectly rational higher being. The theory is illustrated by a medical example.
In the context of a new approach towards quantum foundation, the Born rule is proved under weak conditions through several steps. Some of the steps turn out to have connections to statistical theory. One step gives a generalized likelihood principle, a result of independent interest. The sole additional assumption behind the Born rule is a consequence of the likelihood principle of statistics. This principle is discussed from several points of view. The whole approach towards quantum foundation advocated here, leads to a simpler and more intuitive pure state concept than the traditional one in quantum mechanics. It also provides links to statistical theory and to Andrei Khrennikov's macroscopical quantum-like models, in particular, models connected to cognition and decisions. Finally, the approach has connections to relativity theory and to quantum field theory. It is a hope that this approach now can be seen as a good alternative to the traditional formal foundation of quantum theory. The approach is here discussed in the case of finite-values parameters, but continuous parameters and parameters taking a countable set of values can be included in the theory by approximating them with finite-valued ones.
Quantum decision theory is introduced here, and a new basis for this theory is proposed. It is first based upon the author’s general arguments for the Hilbert space formalism in quantum theory, and next on arguments for the Born rule, which is the basis for calculating quantum probabilities. A basic notion behind the quantum theory foundation is that of theoretical variables, that are divided into accessible and inaccessible ones. This is here specialized to decision variables. It is assumed that each accessible variable can be seen as a function of a specific inaccessible variable. Another assumption is that there exist two maximal accessible decision processes in the given situation. Two basic assumptions behind the Born rule are 1) the likelihood principle, and 2) the actor in question has motivations that can be modeled by a hypothetical perfectly rational higher being. The theory is illustrated by a medical example.
It is argued from several points of view that quantum probabilities might play a role in statistical settings. New approaches toward quantum foundations have postulates that appear to be equally valid in macroscopic settings. One such approach is described here in detail, while two others are briefly sketched. In particular, arguments behind the Born rule, which gives the basis for quantum probabilities, are given. A list of ideas for possible statistical applications of quantum probabilities is provided and discussed. A particular area is machine learning, where there exists substantial literature on links to quantum probability. Here, an idea about model reduction is sketched and is motivated from a quantum probability model. Quantum models can play a role in model reduction, where the partial least squares regression model is a special case. It is shown that for certain experiments, a Bayesian prior given by a quantum probability can be motivated. Quantum decision theory is an emerging discipline that can be motivated by this author’s theory of quantum foundations.
It is argued from several points of view that quantum probabilities might play a role in statistical settings. New approaches toward quantum foundations have postulates that appear to be equally valid in macroscopic settings. One such approach is described here in detail, while one other is briefly sketched. In particular, arguments behind the Born rule, which gives the basis for quantum probabilities, are given. A list of ideas for possible statistical applications of quantum probabilities is provided and discussed. A particular area is machine learning, where there exists substantial literature on links to quantum probability. Here, an idea about model reduction is sketched and is motivated from a quantum probability model. Quantum models can play a role in model reduction, where the partial least squares regression model is a special case. It is shown that for certain experiments, a Bayesian prior given by a quantum probability can be motivated. Quantum decision theory is an emerging discipline that can be motivated by this author's theory of quantum foundations.
A new approach to quantum theory is proposed in this paper. The basis is first taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an actor to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an actor or to some communicating actors. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the discrete case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is shown here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. This simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. To focus only on physical variables rather than mathematical variables, the concept of inaccessible variables is then replaced by the concept of notions, and in this connection, aspects of category theory partly replace group theory. The interpretation inferred from the proposed foundation here may be called a general epistemic interpretation of quantum theory. A special case of this interpretation is QBism; it also has a relationship to several other interpretations.