Quantum-like imprecise probabilities

Abstract The primary contribution of this work is the identification of the equivalence between imprecise probabilities and quantum probabilities, resulting in the development of the concept of probability waves. Probability waves serve as a straightforward mathematical tool for generalizing the most prevalent description of imprecise knowledge, which entails an interval with lower and upper bounds denoted by $$\underline{p}(y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:munder> <mml:mi>p</mml:mi> <mml:mo>̲</mml:mo> </mml:munder> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$\overline{p}(y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>p</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Probability waves establish a connection between imprecise probabilities and contemporary theories of quantum cognition, which elucidate how humans make decisions for uncertain events. The Bayesian interpretation of probability waves facilitates the identification of instances where knowledge of distribution parameters is insufficient to make any decision.