Материал: искусственный интеллект

by using substantive theory from the domain under investigation. We describe this step in more detail in the original articles.

3.5 Compute the Choice Probabilities

Once the projectors have been deÞned, it is straightforward to compute the probabilities for any contingency table using the quantum algorithm described earlier. For example, the probabilities for the AB table are obtained from the equation p(A = i, B = j ) = PB=j PA=i ψ 2, and the probabilities for the A, B, D table are obtained from the equation p(A = i, B = k, D = j ) = PD=j PB=k PA=i · ψ 2.

3.6 Estimate Model Parameters, Compare and Test Models

Once the model has been deÞned, the parameters of the initial state ψ and the parameters in the Hamiltonian matrix H can be estimated from the frequencies contained within contingency table data. This can be accomplished by using maximum likelihood estimation procedures. Suppose the dimension equals n (n = 15 in our example). If we use a real valued initial state, then initial state has n − 1 parameters (because the state is restricted to unit length). If the Hamiltonian is restricted to real values, then the Hamiltonian has (n · (n + 1)/2) − 1 parameters (one diagonal entry is arbitrary). However, often it is possible to use a lower number of parameters for the Hamiltonian. Model comparison methods, such as Bayesian information criterion or Akaike information criterion, can be used to compare models for accuracy adjusted for parsimony (deÞned by number of parameters).

HSM models can also be empirically tested using a generalization criterion. After estimating the projectors from two-way tables shown in Fig. 1, the model can be used to make a priori predictions for table A by D or for a three-way table such as A by B by D. This provides strong empirical tests of the model predictions.

The model also provides interpretable parameters to help understand the complex array of contingency tables. The estimate of the initials state ψ provides the initial tendencies to respond to questions. In the previous example, ψ represents the probabilities to respond to the A, D questions. The rotation, U ψ gives the initial tendencies to respond to the B, C questions. The squared magnitude of an entry in the unitary matrix, uj k 2, represents the squared correlation between a basis vector representing an event in one basis (e.g., an event in the A, D basis) and a basis vector representing an event in another basis (e.g., the B, C basis). These squared correlations describe the inter-relations between the variables, independent of the initial state.

rently located at the following site http://mypage.iu.edu/~jbusemey/quantum/ HilbertSpaceModelPrograms.htm.

The site contains a link to some commonly used programs required for all of the models. It also contains programs designed to Þt (a) collections of one and two-way tables made from binary variables, such as those that appear in [9], (b) one and twoway tables for variables with 2, 3, 4 values, such as those that appear in [8], (c) a model for order effects between a pair of variables with a relatively large (e.g., nine or greater) levels of rating scale values, such as those that appear in [19].

5 Concluding Comments

HSM models provide a simple and low dimensional method for representing multiple contingency tables formed from measurement of subsets of variables. This simple representation in low dimensional spaces is achieved by using ÒrotationÓ of the basis vectors to generate new incompatible variables. Bayesian network models can also be applied to collections of tables; however, these types of models assume the existence of a complete joint distribution of the observed variables, and it is often the case that no complete joint distribution can reproduce the tables because of violations of constraints imposed by marginalization. HSM models can be applied to collections of tables even when no complete joint distribution exists to reproduce the collection. Of course, HSM models do not provide the only way, and there are other probabilistic models that could be considered such as the use of probabilistic data base programming methods [6]. However, HSM models have been shown to provide successful accounts of actual empirical data [8, 9, 17, 19], as well as the possibility for providing new a priori predictions for new data, which is not the case for probabilistic database programming methods.

Acknowledgements This research was based upon the work supported by NSF SES-1560501 and NSF SES-1560554.

References

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Basics of Quantum Theory for Quantum-Like Modeling Information Retrieval

Andrei Khrennikov

Abstract This chapter contains a brief introduction to the mathematical formalism and axiomatics of quantum mechanics (QM). Recently quantum mathematics and methodology started to be widely used for modeling decision making for humans and AI-systems, including quantum-like modeling information retrieval. Experts in such areas do not go deeply into the details of quantum theory. Moreover, typically such consumers of quantum theory do not use all its components. Quantum measurement theory is the most useful for application, including information retrieval. The main issue is the quantum treatment of incompatible observables represented mathematically by noncommuting Hermitian operators. At the level of statistical data incompatibility is represented as interference of probabilities, in the form of modiÞcation of the formula of total probability by adding the interference term.

1 Introduction

Recently the mathematical formalism and methodology of QM, especially theory of quantum measurement, started to be widely applied outside of physics1: to cognition, psychology, economics, Þnances, decision making, AI, game theory, and information retrieval (for the latter, see, e.g., [24, 51Ð53, 56, 58, 59] and the chapters in this book). This chapter contains a brief introduction to the mathematical formalism and axiomatics of QM. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability

1See, for example, [1Ð10, 15, 27, 28, 32, 38Ð41, 43Ð48, 55, 57].

A. Khrennikov ( )

Linnaeus University, International Center for Mathematical Modeling in Physics and Cognitive Sciences, VŠxjš, Sweden

e-mail: andrei.khrennikov@lnu.se

D. Aerts et al. (eds.), Quantum-Like Models for Information Retrieval and Decision-Making, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-25913-6_4

model (Kolmogorov [49]). Then we present basics of theory of Hilbert spaces and Hermitian operators, representation of pure and mixed states by normalized vectors and density operators. This introduction is sufÞcient to formulate the axiomatics of QM in the form of Þve postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We distinguish sharply the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, the von NeumannÕs and LŸdersÕ forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (in LŸdersÕ form) plays the crucial role in the deÞnition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables, as a modiÞcation of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works of Khrennikov (see, e.g., [29Ð38]).

Since classical probability theory is based on the Boolean algebra of events a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, the so-called quantum logic. The latter is also brießy presented in a separate section.

We continue this review with a new portion of Òquantum mathematics,Ó namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on DiracÕs notation with ketand bra-vector, we discuss brießy the notion of qubit and entanglement of a few qubits. This chapter is Þnished with the presentation of the detailed analysis of the probabilistic structure of the two-slit experiment, as a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and the non-Kolmogorovean probabilistic structure of this experiment.

We hope that this chapter would be interesting for newcomers to quantum-like modeling. May be even experts can Þnd something useful, say the treatment of interference of probabilities as a violation of the law of total probability. In any event, this chapter can serve as the mathematical and foundational introduction to other chapters of this book devoted to the concrete applications.

2 Kolmogorov’s Axiomatics of Classical Probability

The main aim of QM is to provide probabilistic predictions on the results of measurements. Moreover, statistics of the measurements of a single quantum observable can be described by classical probability theory. In this section we shall present an elementary introduction to this theory.

We remark that classical probability theory is coupled to experiment in the following way: