The Mathematical Rule that Makes Sense of Illusions

Tricks of the mind.

Posted Feb 22, 2021

When Hermann von Helmholtz was a small child in early nineteenth-century Prussia, he walked through his hometown of Potsdam with his mother. Passing a stand containing small dolls aligned in a row, he asked her to reach out and get one for him. His mother didn’t oblige him, however, because she couldn’t. There were no dolls. What the young Helmholtz was experiencing was an illusion; the ‘dolls’ he saw near him were actually people far away, at the top of the town’s church tower.

While this may seem a silly mistake for the child to make, we are all prone to optical illusions. While Helmholtz thought large people were small dolls, you may think the two figures in this famous example of the Ponzo illusion are different sizes when they’re actually the same:

Used with Permission of Grace Lindsay
Source: Used with Permission of Grace Lindsay

Why do our brains do this? Is it just a flaw in their wiring? Or is there a good reason behind it? Helmholtz---who went on to become an influential physician, physiologist, and physicist---had a hunch. He concluded that perception is not just a matter of what’s in front of the eyes. Instead, a large amount of processing must go on between the point at which sensory information comes in and the moment it becomes a conscious experience. The result of this processing, he wrote, is ‘equivalent to a conclusion, to the extent that the observed action on our senses enables us to form an idea as to the possible cause of this action’.

Helmholtz’s intuition was a reasonable one and has influenced thought in psychology for decades. But to truly test a hypothesis like Helmholtz’s, it has to be formalized; specifically, it has to go from mere words to an explicit---that is, mathematical---model. Mathematical models are a way to describe a theory about how the brain works precisely enough to communicate it to others and to predict the outcomes of future experiments. By building mathematical models, scientists can codify and then test their intuitions.

Starting in the 1990s, psychologists did just this. The mathematical tool that realized Helmholtz’s hunch is known as Bayes’ Rule. Actually discovered by Pierre-Simon Laplace in 1812 but named after English minister Thomas Bayes, Bayes’ rule says to come to a conclusion we need to combine two different sources of knowledge. 

The first is known as the “prior” and it represents understanding that has accumulated over time. For example, the young Helmholtz may have become very accustomed to seeing dolls and so his prior would say that the presence of dolls is very likely. The second source is the “likelihood”, which is the evidence you have here and now. In Helmholtz’s case, this is the impression he got from the light that hit his eyes as he looked up at that church tower. Bayes’ Rule says we can multiply these terms together to get a “posterior”, which provides the probability of any given conclusion. For Helmholtz, perhaps a cloudy day made the evidence from his eyes less clear and an obsession with toys made his prior very strong, and these combined to tell him the most probable conclusion was that dolls were in that town square.     

Can Bayes’ Rule account for illusions in a scientifically rigorous way? In 2002, a team of researchers cataloged a series of common illusions that people fall prey to when trying to estimate the movement of an object. It included the fact that the shape of an object influences the direction we think it is moving in, that two items moving in different directions may appear as one and that dimmer objects appear to move more slowly. This may seem simply like a list of our failings, but the researchers found that all of these lapses could be explained by a simple Bayesian model. 

Particularly, these habits fall out of the calculation if we assume a specific prior: that motion is more likely to be slow than fast. Take, for example, the last illusion. When an object is hard to see, the evidence it provides about its movement is weak. In the absence of evidence, Bayes ’ rule relies on

the prior – and the prior says things move slowly. This bit of mathematics may explain why drivers have a tendency to speed up in the fog – with weak information about their own movement, they assume their speed is too slow. 

Importantly, the Bayesian approach recasts these tricks of the mind as traits of a rational calculation. It shows how some mistakes are actually reasonable guesses in an uncertain world.    

This blog post is adapted from the book Models of the Mind: How Physics, Engineering, and Mathematics Has Shaped Our Understanding of the Brain (Bloomsbury Sigma).