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Non-Commutative Probability Models in Human Decision Making: Binary Hypothesis Testing

Aneesh Raghavan, John S. Baras

Published Jan 1, 2019• 3 citations
Publisher

Abstract

In this paper, we consider the binary hypothesis testing problem, as the simplest human decision making problem, using a von-Neumann non-commutative probability framework. We present two approaches to this decision making problem. In the first approach, we represent the available data as coming from measurements modeled via projection valued measures (PVM) and retrieve the results of the underlying detection problem solved using classical probability models. In the second approach, we represent the measurements using positive operator valued measures (POVM). We prove that the minimum probability of error achieved in the second approach is the same as in the first approach.

DOI
10.1016/j.ifacol.2019.01.019

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We study two relevant characterizations of a commutative positive operator valued measure (POVM) F . The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ . We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ . Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.

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In this paper, we consider the binary hypothesis testing problem, as the simplest human decision making problem, using a von-Neumann non-commutative probability framework. We present two approaches to this decision making problem. In the first approach, we represent the available data as coming from measurements modeled via projection valued measures (PVM) and retrieve the results of the underlying detection problem solved using classical probability models. In the second approach, we represent the measurements using positive operator valued measures (POVM). We prove that the minimum probability of error achieved in the second approach is the same as in the first approach.
Abstract In this chapter we first review projective or von Neumann measurements from the point of view of the projection postulate and the Lüders rule, before turning to more general measurements modelled using positive operator-valued measures (POVMs). After examining their mathematical description, we discuss symmetric informationally complete (SIC) POVMs and Zauner’s conjecture, as well as the representation of POVMs as projective measurements on a larger Hilbert space, a result called Naimark dilation. We then a model for non-ideal projective measurements and analyse the properties of such measurements in relation to the properties unbiasedness, faithfulness, and non-invasiveness of ideal projective measurements. Finally, we consider the problem of distinguishing non-orthogonal quantum states and examine how the use of POVMs can be beneficial for this task.
The authors present the theory of quantum measurements in a humanities friendly way. The most general process of decision-making is represented with the aid of the formalism of quantum apparatuses and instruments. This measurement formalism generalizes the standard one based on the von Neumann–Lüders projection postulate. Generalized quantum observables are mathematically represented as positive operator valued measures (POVMs) and state transformers resulting from the feedback of measurements to the states of systems that are given by quantum instruments. The quantum scheme of indirect measurements (a special realization of quantum instruments) is applied to model decision-making as resulting from the interaction between the belief and decision states. The authors analyze the specific features of quantum instruments which are important for cognitive and social applications. In particular, the state transformers given by quantum instruments are in general less invasive than the state projections. Thus quantum-like decision-making need not be viewed as a kind of state collapse. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
FINDING: Quantum probability replaces classical Kolmogorov probability in modeling human decision-making under uncertainty and conflict, with non-commutative operators capturing context-dependent judgment. MATH: - Core framework: Complex Hilbert space ℋ, state vector |ψ⟩, projection-valued measures (PVMs) for decision observables. - Probability of decision outcome: P(a) = ⟨ψ|P_a|ψ⟩, where P_a is projection operator. - Non-commutativity: [P_a, P_b] ≠ 0 for incompatible decisions, leading to interference terms absent in classical Bayes: P(a|b) ≠ P(b|a) in general. - Quantum interference: P(a ∨ b) = P(a) + P(b) + 2√(P(a)P(b)) cos(θ), where θ is phase angle. - No explicit constants (0.382, 0.618, 1.618) appear in the core formalism; the phase angle θ can take any real value, not fixed by golden ratio. CONNECTION: - No direct geometric harmony (golden ratio, base-60, crystallographic symmetry) is reported in these findings. The quantum probability framework is structurally ana Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
In the heart of every physical theory lies measurement theory. While classical measurements are passive probes, QMs can be viewed as active physical processes that shape the available information. Depending on the measurement type, different aspects of the information about a system are retrieved. All possible measurements in quantum physics are described by mathematical objects called Positive Operator-Valued Measures (POVMs). The most familiar standard measurements are called Projective-Valued Measures (PVMs) and make up a subset of POVMs. The complementary subset of POVMs is non-projective. However, some non-projective POVMs can be expressed as a probabilistic mixture of PVMs. Such POVMs are called projectively simulable, while non-simulable POVMs are genuinely non-projective. These can be made projectively simulable by adding noise to the measurements. If a genuinely non-projective POVM requires a large amount of noise before it becomes projectively simulable, it is more non-projective than a POVM that requires less noise. Hence, the degree of non-projectiveness is determined by the required noise strength. The act of measuring a quantum system changes the quantum state. While quantum measurements only take into account the readable classical output value, quantum instruments account for both the classical outcome and the post-measurement state. Hence, instruments are associated with measurements, and in an analogous way, projective and projectively simulable instruments can be defined. For many purposes, non-projective POVMs are more useful than PVMs, and the more non-projective a POVM is, the greater the advantage of using them over PVMs. Therefore, the aim of the thesis is to investigate the genuinely non-projective POVMs. More precisely, the degree of non-projectiveness in qubit POVMs is studied. This is done by convex optimization, specifically by solving semidefinite programs. Furthermore, the thesis addresses the open problem of finding the most non-projective qubit quantum instrument by a witness-based see-saw algorithm.
We study two relevant characterizations of a commutative positive operator valued measure (POVM) F . The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ . We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ . Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.