Coherent Upper Conditional Previsions with Respect to Outer Hausdorff Measures and the Mathematical Representation of the Selective Attention

Coherent upper conditional probabilities defined by Hausdorff outer measure are proposed to represent the unconscious activities of the human brain when information are given. In the model uncertainty measures are defined according to the complexity of the conditioning event that represent the given information. The model is applied to explain mathematically 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.