Visual illusions : optimal inferences ?

Reconciling the work of Kahneman and Tversky with the numerous works that support the Bayesian perspective is one of the major areas of research over the next few years. In the lecture, without resolving this apparent contradiction, we touched on a few avenues of research. First of all, it should be noted that many of the difficulties observed by Kahneman and Tversky do not necessarily stem from the evaluation of probabilities, but rather from decision-making. The utility function, but also the consideration of risk, could explain some of the observed biases. Secondly, the brain, which is not a digital computer, may only implement an imperfect Bayesian algorithm. It could be that the representation of very small probabilities, or the multiplication of two probabilities or two distributions, poses particular difficulties for the brain.

Thirdly, Kahneman and Tversky use conscious, verbal, high-level tasks. However, our conscious workspace takes into account very little data: conscious processing operates in a serial and fundamentally limited mode (see 2009-2010 lecture). On the contrary, it could be that the non-conscious processing underlying perception, motor decisions, language learning and so on, calls upon massively parallel algorithms, closer to Bayesian optimality. A recent line of research involves comparing problems which are formally identical in mathematical terms, but which are solved either by conscious verbal reflection, or unconsciously by the sensorimotor system. Indeed, very different biases are observed (Braun, Nagengast & Wolpert, 2011).

Finally, the context and words used to present a statistical problem can have an important effect. Our brains may process certain types of problem more easily, either because they spontaneously call upon areas of knowledge for which evolution has endowed us with specific solutions (Cosmides, 1989), or because the format in which the problem data is presented is not easy to recode into the internal format we use for Bayesian inference (Gigerenzer & Hoffrage, 1995). Gigerenzer thus challenges the work of Kahneman and Tversky by showing, for example, that the a priori probability neglect bias largely disappears when the problem is formulated in the form of what he calls "natural frequencies", i.e. with whole numbers.

Krynski and Tenenbaum (2007) propose a separate, equally interesting hypothesis. In their view, the appropriate theoretical framework for Kahneman and Tversky's "judgments under uncertainty" is not traditional statistical inference, but causal inference. Our brains would not have evolved to compile statistics on complex conditional probabilities P(A|B,C,D...) (a problem which, incidentally, quickly becomes impossible to evaluate mathematically), but rather to draw inferences from causal models of the observations made. The interpretation of a statistical problem would therefore involve three stages: (1) construction of a mental model of causal relationships, (2) assignment of a value to the various parameters of the model, (3) Bayesian inference. The construction of an inappropriate causal model would explain the subjects' errors. Krynski and Tenenbaum tested their hypothesis by showing that changing one sentence in the problem statement was enough to change the causal model used by the participants. Immediately, inference capacity improved: even in the absence of precise numerical values, the test subjects showed correct intuitions of the probabilities involved.

It would seem, then, contra Kahneman and Tversky, that the human brain understands the intuitive and approximate mechanisms of Bayesian inference. However, these are only deployed appropriately if the presentation of the problem leads the public to imagine the appropriate causal model.

Visual illusions: optimal inferences?

Visual perception provides an excellent example of the non-conscious application of Bayesian statistics. As early as 1867, in his famous Treatise on Physiological Optics(Handbuch der physiologischen Optik), Hermann von Helmholtz introduced the concept of "unconscious inference": sensory inputs (" Perception ") would combine with our expectations (" Vorstellung ") to form the content of conscious experience (" Anschauung ").

Indeed, the notion of inference applies well to the problem posed by perception. Indeed, all our sensory inputs are ambiguous - each of them could result from an infinite number of possible causes. The task of our perceptual system is therefore to select the most plausible solution from an infinite number of possible solutions. Bayesian theory explains this optimal choice process on the basis of (1) a priori knowledge of the probabilities of objects in the external world p(O) (which may result from an accumulation of knowledge during learning), (2) knowledge of the likelihood function p(S|O) (which may result from an internal model of object behavior), and (3) application of Bayes' rule: p(O|S) α p(S|O) p(O).

According to this theory, many visual illusions can be explained by an optimal inference process that takes into account the statistics of the environment. If perception is optimal, why does it generate an illusion? Because the stimulus that creates the illusion is very impoverished compared to our usual experience of the outside world. The perceptual system then enriches it with statistical information accumulated from past experience. The integration ofa priori and sensory inputs results in a conscious perception that deviates from actual sensory data. However, although it thus leads to an illusion, it is not inappropriate to call this perceptual inference "rational", insofar as, in a more natural context, it would be useful or even optimal for the organism.

Simoncelli and colleagues analyze biases in the perception of a bar's orientation from this perspective (Girshick, Landy & Simoncelli, 2011). Perception varies with orientation, both in its accuracy (internal variability is minimal for vertical and horizontal) and in its bias (responses are biased towards horizontal and vertical orientations). Theoretical analysis shows that these data can be fully explained, within the Bayesian framework, by thea priori higher frequency of these cardinal directions in the outside world.

Motion illusions succumb to a similar interpretation (Weiss, Simoncelli & Adelson, 2002). From a related theoretical framework, Dale Purves and colleagues analyze many other visual and auditory illusions, showing that they can be explained by an internalization of statistics from the external world. To take just one example, because of perspective, a vertical bar on the retina often reflects a greater distance in the real world than a horizontal bar - which is why, according to Purves, it is also perceived as longer (Howe & Purves, 2002; Purves, Lotto, Williams, Nundy & Yang, 2001).