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5. Modeling of the process of sensation–perception within indirect measurement scheme

Foundations of theory of unconscious inference for the formation of visual impressions were set in 19th century by H. von Helmholtz. Although von Helmholtz studied mainly visual sensation–perception, he also applied his theory for other senses up to culmination in theory of social unconscious inference. By von Helmholtz here are two stages of the cognitive process, and they discriminate between sensation and perception as follows:

• Sensation is a signal which the brain interprets as a sound or visual image, etc.
• Perception is something to be interpreted as a preference or selective attention, etc.

In the scheme of indirect measurement, sensations represent the states of the sensation system  of human and the perception system plays the role of the measurement apparatus . The unitary operator  describes the process of interaction between the sensation and perception states. This quantum modeling of the process of sensation–perception was presented in paper (Khrennikov, 2015) with application to bistable perception and experimental data from article (Asano et al., 2014).

6. Modeling of cognitive effects

In cognitive and social science, the following opinion pool is known as the basic example of the order effect. This is the Clinton–Gore opinion pool (Moore, 2002). In this experiment, American citizens were asked one question at a time, e.g.,

$A=$ “Is Bill Clinton honest and trustworthy?”

$B=$ “Is Al Gore honest and trustworthy?”

Two sequential probability distributions were calculated on the basis of the experimental statistical data, $p_{A,B}$ and $p_{B,A}$ (first question$A$  and then question $B$ and vice verse).

6.1. Order effect for sequential questioning

The statistical data from this experiment demonstrated the question order effect QOE, dependence of sequential joint probability distribution for answers to the questions on their order $p_{(A,B)}\neq p_{(B,A)}$. We remark that in the CP-model these probability distributions coincide:

$p_{A,B}(\alpha ,\beta )=P(\omega \in \Omega :A(\omega )=\alpha ,B(\omega )=\beta )=p_{A,B}(\beta ,\alpha )$

where $\Omega$ is a sample space $P$ and  is a probability measure.

QOE stimulates application of the QP-calculus to cognition, see paper (Wang and Busemeyer, 2013). The authors of this paper stressed that noncommutative feature of joint probabilities can be modeled by using noncommutativity of incompatible quantum observables  $A,B$ represented by Hermitian operators ${\widehat {A}},{\widehat {B}}$ . Observable  $A$ represents the Clinton-question and observable $B$ represents Gore-question. In this model, QOE is identical incompatibility–noncommutativity of observables:

$[{\widehat {A}},{\widehat {B}}]\neq 0$