In the 18th century, there was a very interesting non-conformist priest called Bayes, Thomas Bayes, B-A-Y-E-S, who was an amateur mathematician, and he came up with a formula for relating the probability of the evidence to the probability of the previous knowledge. So he called this priors, as previous knowledge, posteriori, the evidence added on by the particular observation and the formula relates to these in a mathematical way. This became important for, as I understand it, for decision-making in economics and economists have mathematical models for making decisions based on Bayesian statistics, which incidentally he never published in his own lifetime. A friend found this paper after he was dead, you know, put it to the Royal Society and then it was published and it’s now one of the most famous papers, I suppose, certainly in that area of science that there is. Incredibly important, actually, because it’s quantified what was a rather nebulous idea, I think, in a way, following on from Ockham, you could say. Anyway, what I find interesting is looking at the Bayesian formulation both for science, that is hypothesis in science, and then for perceptions because I’ve always thought that perceptions are hypothesis. I think the brain is making a guess at what is out there on essentially inadequate evidence, it’s doing the best it can, and it’s always got the probability of the available evidence in terms of the hypothesis to be tested, and, of course, the more unlikely the hypothesis, the more strong you need the probabilities of the evidence in order to modify it or change it or throw it out. And it may throw it out wrongly; it may pop back again later. So when you get these ambiguities in perception, like the duck rabbit, that is entertaining alternatives because there’s no clear Bayesian imbalance in the probabilities between different hypothesis so it oscillates, it modifies the system such that the duck is seen the rabbit is seen. Now, what I like about the hollow face illusion is this. This is a hollow mould of a face, so the nose is actually sticking in instead of out but, blow me down, you look at this thing from a bit of a distance and it absolutely looks like an ordinary face with the nose sticking it, convex. Why? Because of Bayesian probabilities. The brain says to itself, that’s ridiculous, this thing is clearly a face, it’s got eyes, it’s got a mouth, etc, how on earth can the thing be hollow when it’s a face because faces are not hollow things. So the prior probability dominates, throws out the evidence of the senses, the eye, and the illusion dominates. So it’s not like the duck rabbit, this could be either. The illusion, the false possibility, dominates over the truth because it is unlikely, from past experience, that the face is hollow. This is the point. Now, this is the sort of Bayesian strategy and what’s interesting about Bayes is that he actually quantified this, set up a really neat mathematical formulation and so you could put this into a computer, if you like, you can treat it objectively, explicitly, with the methods of science, and yet, looking at the other side of it, artists are playing about with this all the time, you know, presenting shapes which might be one thing, might be another, they sort of morph in the brain from one thing to another so the artist plays about with what in science you can define in terms of Bayesian statistics. To add just a little bit to this, the statistics here are subjective, the probabilities are subjective, they’re in your brain. The reality provides relative frequencies of events like how often it rains on a Thursday, or how often it rains in summer or winter, that’s objective, but the probability assigned to that is given by the creative, intelligent, or not so intelligent, brain. So you’ve got this wonderful interplay between objectivity of events, frequencies of events, and the subjective assessment of the probabilities which can be wildly different from the ratio of frequencies in the external world objectively so that the subject and the object can be way apart and artists and scientists, I think, are playing with opposite extremes of this.