Hard to believe it’s been eight years since we last touched upon Bayesian models, and they are just as relevant as ever. A delightful new book Models of the Mind, authored by computational neuroscientist Grace Lindsay, devotes a chapter to how probability and Bayes’ rule helps people make rational decisions. It was our old acquaintance and intellectual mentor, Hermann von Helmholtz, who seriously pondered the question of how perception can arise out of ambiguous or uncertain inputs. His ideas became known as “unconscious inference”, and this ultimately became the basis for modern day Bayesianism.
The easiest way to depict Bayes’ Rule is as follows:
The mathematical form of Bayes’ Rule is:
P(h I d) = P(d I h)P(h)/P(d), where “h” represents the hypothesis and “d” the observed data. The term on the left-hand side of the equation is known as the posterior distribution. Bayesian decision theory (BDT) addresses how Bayes’ rule guides decisions by indicating how the posterior distribution should be mapped on to a specific perception, choice or action. Each perception involves a bit of computation according to Bayes’ rule.
Using movement experiments in the laboratory as an example Lindsay notes that for the most part, humans behave as good Bayesians. When the visual evidence is weak, we depend more on our vestibular system. However even when visual information is more reilable, we still don’t use vision as much as Bayes’ rule would predict. Vestibular input is consistently over-represented. This could be a result of the fact that the visual input is always a bit ambiguous as compared to vestibular.
In 1993, a group of researchers met in Chatham, Massachusetts for a working symposium to probe how prior knowledge is brought to bear upon the interpretation of sensory data, and in particular within the visual system. The meeting gave birth to a book edited by David Knill and Whitman Richards in 1996 titled Perception as Bayesian Inference.
You may be familiar with the book’s co-editor, Whitman Richards from MIT, for his seminal paper on stereopsis and stereoblindness published in Experimental Brain Research in 1970. As it turns out, 25 years after publication of the book the Bayesian approach remains a hot topic. Put the word “Bayesian” in the search box of the Journal of Vision and you’ll come up with a staggering 3702 entries!
If you’re looking for a primer on how Bayesian modeling relates to visual perception, you can do no better than this article by Mamassian, Landy and Maloney. It covers the gamut, from mutli-stable percepts to stereoscopic vision.