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

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 rst 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 rst approach.