Coherent conditional previsions with respect to inner and outer Hausdorff measures to represent conscious and unconscious human brain activity

Abstract

Coherent upper and lower conditional probabilities defined by Hausdorff outer and inner measures are proposed to represent respectively the unconscious and conscious activities of the human brain when information is given. In the model uncertainty measures are defined according to the complexity of the conditioning event that represents the given information. The model is applied to explain mathematically the conjunction fallacy in Linda's Problem and the bias of selective attention described in the so-called invisible Gorilla experiment, that is often taken as a characteristic example of the inescapable limitations of human perception. Once people are concentrated on doing a specific action, they do not notice unexpected events (having 0 probability) occurring in the meantime. When applying the model, selective attention is no longer a bias since it is able to explain this function of the human brain mathematically. Moreover different reactions of people to unexpected events can be represented in different metric spaces with metrics which are not bi-Lipschitz. In these metric spaces coherent upper conditional probabilities defined by Hausdorff outer measures are not mutually absolutely continuous and so they do not share the same null events.