A Model of Sensory Perception in Hilbert Space

Baldo, Marcus (2006) A Model of Sensory Perception in Hilbert Space. In: 10th annual meeting of the Association for the Scientific Study of Consciousness, 23-26 Jun 2006, Oxford, UK.

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Abstract

Conscious perception often involves, beyond the detection of a given sensory stimulus, its comparison with other concurrent stimuli as well, encompassing tasks akin to discrimination, identification and categorization measures. Over the last forty years, Signal Detection Theory (SDT) has been the paradigmatic source of psychophysical models for perceptual processing. Here I offer a conceptual framework based on the vector structure of Hilbert spaces that, besides assimilating in a natural way the probabilistic character of perceptual operations as formulated in classical SDT, avoids some of its inherent drawbacks. Its essence is the representation of a percept by a vector or, more generally, by a subspace of a vector space S. A sensory stimulus, being initially broken down into its components, is further transformed yielding the generation of a normalized vector representing a sensory state in a Hilbert space. A given class of perceptual outcomes would be thus equivalent to an observable specified by an operator H. For a system in a sensory state s, the probability prob(H, p) that the perceptual processing will result in the percept p is given by the scalar product between s and Us, where U is the projection operator onto the ray containing v, an eigenvector of H whose p is the corresponding eigenvalue. This statistical algorithm relates a set of perceptual outcomes to the probabilities of their occurrence. The present model, while quantitatively consistent with the main results predicted by standard SDT, offers a more straightforward computational tool for measures of detectability and response bias. In addition, it is able to capture, in a more natural and unifying perspective, some perceptual phenomena usually left out by the conventional theory. The novel theoretical framework presented here is substantiated by empirical data stemming from temporal order judgments and spatial localization tasks performed by human volunteers.

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