Tuesday, April 01, 2008

Bayesianism Finds Another Home (Outside of the Law) -- but (allegedly) Runs into Difficulties

Dale Purves, M.D., Center for Cognitive Neuroscience, Duke University, writes:
Information in visual stimuli cannot be mapped unambiguously back onto real-world sources, a quandary referred to as the "inverse optics problem." The same problem exists in all other sensory modalities.
In the case of visual perception, Dr. Purvis takes a "probabilistic approach." His approach also emphasizes the role of experience. He writes:
A Primer on Probabilistic Approaches to Visual Perception

A growing body of evidence indicates that visual percepts are generated according to the empirical significance of light stimuli, rather than the characteristics of the stimuli as such. ...

Background

The first and most influential advocate of using past experience as a means of contending with the uncertain provenance of visual stimuli was Hermann von Helmholtz (1866/1924). Helmholtz summarized his conception of this empirical contribution to visual percepts by proposing that the raw "sensations" generated by the physiological infrastructure of the eye and the input stages of the visual brain could be modified by information derived from experience. Helmholtz described this process as making "unconscious inferences" about reality, thus generating perceptions more nearly aligned with stimulus sources when input-level sensations proved inadequate (op cit., vol. III, p.10 ff). Despite these speculations and the ensuing debate during the second half of the 19th C., vision science during most of the 20th C. has been understandably dominated by the enormous success of modern neurophysiology and neuroanatomy. A plausible assumption in much contemporary vision research has thus been that understanding visual perception will be best achieved by gleaning increasingly precise information about the receptive field properties of visual neurons and the synaptic connectivity that gives rise to these properties. As a result, the role of past experience in determining what observers see has, until recently, received relatively little attention.

Bayes' theorem

If the visual system uses empirical information to generate perceptions that reflect the real-world conditions and object relationships that observers have always had to respond to by appropriate visually-guided behavior, then understanding vision inevitably means understanding how, in statistical terms, physical sources are related to retinal images. By far the most popular approach to meeting this challenge has been Bayesian decision theory. ... In vision research, Bayes' theorem was initially used to develop pattern recognition strategies for computer vision. ... More recently, however, the framework provided by the theorem has been advocated as a means of rationalizing visual perception (or at least the judgments associated with visual perception). ... With respect to vision, the prior describes the relative probabilities of different physical states of the world pertinent to retinal images, i.e., the relative frequency of occurrence of various illuminants, surface reflectance values, object sizes and so on. The second term, P(E|H), is called the likelihood function. If hypothesis H were true, this term indicates the probability that the evidence E would have been available to support it. In the context of vision, given a particular state of the physical world (i.e., a particular combination of illumination, reflectance properties, object sizes etc.), the likelihood function describes the probability that the state would generate the retinal projection in question. The product of the prior and the likelihood function, divided by a normalization constant, P(E), gives the posterior probability distribution, P(H|E). The posterior distribution defines the probability of hypothesis H being true, given the evidence E. In vision, the posterior probability distribution thus indicates the relative probability of a given retinal image having been generated by one or another of the different physical realities that might be the source of the image.

...

Because the posterior distribution indicates only the relative probabilities of a set of possible image sources, a particular source (i.e., a particular combination of illumination and reflectance in the example above) must be selected from this set if the aim is to predict what an observer will actually see. The usual way of addressing this further issue is to assume that the visual system makes this choice according to the behavioral consequences associated with each perceptual "decision". The influence of various consequences is typically expressed in terms of the discrepancy between the decision made and the actual state of the world, which over the full range of the possible choices defines a gain-loss function. Since there is no a priori way to model this function (indeed, given the enormous number of variables involved, a realistic gain-loss function for some aspect of vision would be extraordinarily difficult to determine), the relative cost of different behavioral responses is assumed. ...

...

In sum, Bayesian decision theory determines the physical source(s) capable of generating a given retinal image and the relative probabilities of their actually having done so; the percepts predicted are therefore explicit models of world structure.

Empirical ranking theory

The application of Bayesian decision theory to vision is clearly an important advance in that it formalizes Helmholtz's general proposal about "visual inferences" as a means of contending with stimulus uncertainty. Nonetheless, its implementation presents both conceptual and practical difficulties. With respect to the conceptual implications of Bayesian theory applied to visual perception, the intuitively appealing idea that percepts correspond to physical characteristics such as surface reflectance is problematic and in many instances false (as we explain in a later section). Practical obstacles are the difficulty determining the physical parameters relevant to any specific prior, and the need for a decision rule based on an assumed gain-loss function. Is there, then, any other way of conceptualizing how vision utilizes empirical information to deal with the inverse optics problem?

The alternative to Bayesian decision theory that we have used in rationalizing visual percepts begins by abandoning the idea that vision entails inferences (whether conscious or unconscious) about the properties of the physical world, the concept inherent in the application of Bayes' theorem to visual perception. The conceptual basis of this alternative approach is that the percept elicited by any particular stimulus parameter (e.g., the brightness elicited by the luminance of a stimulus) corresponds not to a statistically determined value of the relevant qualities in the physical world (e.g., the most likely illumination and reflectance values underlying that luminance), but rather to the relative frequency of occurrence of that particular stimulus parameter in relation to all other instances of that parameter experienced in the past. For example, with respect to the perceptual quality of brightness, the brightness perceived in response to the luminance of region of a visual scene would be determined by how often the specific luminance had occurred relative to the occurrence of all the other luminance values in that context in the past experience of observers. In other words, the brightness elicited by a target is determined by the empirical rank of the relevant luminance value within the full range of experience with similar scenes ... . This biological rationale of this approach is that it is obviously desirable to have the full perceptual range for any visual quality (from the brightest percept we can have to the dimmest, for example) aligned with the full range of the relevant stimulus parameters generated by the physical world (from the most intense luminance experienced in visual stimuli to the least intense). ...

It should be apparent from this account that the fundamental difference between these two empirical approaches - Bayesian decision theory and the empirical ranking theory - is their different conception of visual perception. Bayesian decision theory, as it has typically been applied to vision, supposes that perceptions are effectively inferences about physical properties of the objects and conditions in the world. In empirical ranking theory visual perceptions are conceived as statistical constructs that have no direct correspondence to the possible real-world sources of a stimulus. In this alternative framework, visual percepts are simply subjective sensations that link visual stimuli to the empirical significance of their sources according to the success or failure of visually guided behavior in past experience. Deciding which approach is the more useful and the more appropriate framework for predicting and understanding will depend on the ability of these theories to explain the full range of the numerous puzzles vision presents.

References

...

Yang, Z, Purves, D (2004). The statistical structure of natural light patterns determines perceived light intensity. Proceedings of the National Academy of Sciences of the United States of America, 101, 8745-8750.

Howe CQ, Purves D (2005) Perceiving Geometry: Geometrical Illusions Explained in Terms of Natural Scene Statistics. New York: Springer.

Catherine CQ, Lotto RB, Purves D (2006) Empirical approaches to understanding visual perception. J Theor Biol 241: 866-875.

I still think that perception involves "unconscious inference" (but my guess is that Bayesian logic and Bayesian statistics are not sufficient to explain how perception works).
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