Generalized Probability Measures and the Framework of Effects

Stanley Gudder

Andrei Yu. Khrennikov, Masanao Ozawa, Felix Benninger +1 more

The past few years have seen a surge in the application of quantum theory methodologies and quantum-like modeling in fields such as cognition, psychology, and decision-making. Despite the success of this approach in explaining various psychological phenomena — such as order, conjunction, disjunction, and response replicability effects — there remains a potential dissatisfaction due to its lack of clear connection to neurophysiological processes in the brain. Currently, it remains a phenomenological approach.In this paper, we develop a quantum-like representation of networks of communicating neurons. This representation is not based on standard quantum theory but on {\it generalized probability theory} (GPT), with a focus on the operational measurement framework. Specifically, we use a version of GPT that relies on ordered linear state spaces rather than the traditional complex Hilbert spaces. A network of communicating neurons is modeled as a weighted directed graph, which is encoded by its weight matrix. The state space of these weight matrices is embedded within the GPT framework, incorporating effect-observables and state updates within the theory of measurement instruments — a critical aspect of this model. This GPT-based approach successfully reproduces key quantum-like effects, such as order, non-repeatability, and disjunction effects (commonly associated with decision interference). Moreover, this framework supports quantum-like modeling in medical diagnostics for neurological conditions such as depression and epilepsy.While this paper focuses primarily on cognition and neuronal networks, the proposed formalism and methodology can be directly applied to a wide range of biological and social networks.

Jonathan Steinberg

Andrei Yu. Khrennikov, Masanao Ozawa, Felix Benninger +1 more

The past few years have seen a surge in the application of quantum theory methodologies and quantum-like modeling in fields such as cognition, psychology, and decision-making. Despite the success of this approach in explaining various psychological phenomena — such as order, conjunction, disjunction, and response replicability effects — there remains a potential dissatisfaction due to its lack of clear connection to neurophysiological processes in the brain. Currently, it remains a phenomenological approach.In this paper, we develop a quantum-like representation of networks of communicating neurons. This representation is not based on standard quantum theory but on {\it generalized probability theory} (GPT), with a focus on the operational measurement framework. Specifically, we use a version of GPT that relies on ordered linear state spaces rather than the traditional complex Hilbert spaces. A network of communicating neurons is modeled as a weighted directed graph, which is encoded by its weight matrix. The state space of these weight matrices is embedded within the GPT framework, incorporating effect-observables and state updates within the theory of measurement instruments — a critical aspect of this model. This GPT-based approach successfully reproduces key quantum-like effects, such as order, non-repeatability, and disjunction effects (commonly associated with decision interference). Moreover, this framework supports quantum-like modeling in medical diagnostics for neurological conditions such as depression and epilepsy.While this paper focuses primarily on cognition and neuronal networks, the proposed formalism and methodology can be directly applied to a wide range of biological and social networks.

Michael Dickson

Abstract This chapter elucidates quantum probabilities in the formal apparatus of quantum mechanics, while not taking sides about interpretive issues. A central contention is that the notion of a probability measure should not be captured in the usual classical framework, but instead in the more general framework of effect algebras. Two formulations of quantum mechanics in this framework are provided: one using projection-valued measures, another using positive-operator-valued measures. The chapter presents several versions of the Born Rule and critically examines justifications that have been proposed for the Born Rule. The chapter concludes with laying down the perennial puzzles about probability in quantum mechanics such as quantum correlations.