Today when people talk about inference, they're really talking about what's referred to as Bayesian inference. And I need to take a few minutes to explain this to you, because it's not simple, but it's very much the concept that people use today to think about vision and visual perception as being inferential. That you're using your experience to guide what you see perceptually based on either the poor quality of the visual image, or its lack of connection to reality as indicated by the inverse problem. So who was Thomas Bayes? Thomas Bayes, as you see here, was about a century before and he was actually primarily a minister, but he was also an amateur mathematician. He came up with Bayes's theorem which is indicated here, and it's a simple statement of conditional probabilities. And why Bayes was interested in this is not really clear historically, but people have used Bayesian decision theory, as is called these days based on Bayes' Theorem. To deal with any situation in which you have an uncertain body of evidence and you're trying to figure out rationally what to do. So first of all, let's go through what Bayes' Theorem is. So on one side of the equation is the probability of A given B, and that's equated to the probability of B given A. And the probability of A in the first place. Well that sounds a little complicated. This side of the equation is called the posterior. These two terms on the right side of the equation are called the likelihood and the prior. All of that makes it seem more complicated than it really is. It's just a statement of conditional probabilities. If you have A and B and you want to work out the rational probabilities that are involved, you can use Bayes's theorem to do it. Well how is that useful? Bayes' theorem is widely used today in a range of applications. Let's take clinical medicine as an example. So let's say you have a test that has false positives. That's an uncertain body of information, and you're trying to decide how rationally to recommend a course of therapy to your patient. So one example would be a mammogram. A mammogram has a lot of false positives. Let's say there are 10 or 20% false positives which I think is probably about right. Where the radiologist sees something it might be a cancer but it might not be, that's the false positive it looks like it might be a cancer but it really wasn't. So what's the rational basis on which you can recommend to a patient the likelihood that it's cancer, that they should have a biopsy which is not entirely innocuous to test. Well, you use Bayes' Theorem to determine that. What is the likelihood of having cancer in a positive mammogram when you know there are a lot of false positives in a series of mammograms. Well, that's just one example. And there are many that indicate the uses of Bayes' Theorem in clinical medicine. In vision A is an image in a stimulus, so this is an image, and B, is the underlying state of the physical world. So A, if you have this image, what's the probability of that image having been created by an underlying state of the world, B? That's the gist of Bayes' Theorem and it's potentially very useful. Again, with uncertainty you can say well what is the likelihood that a given image arose from a given state of the world. So that is attractive to mathematically-minded psychologists and mathematicians interested in vision. And it's been very widely used to explain how vision could be informed by Bayesian decision theory to get around the problem that we've talked about. Not only the false positives, but the inverse problem that we've discussed many times now. How valid can this general idea be as a workaround for the inverse problem? And let's just be reminded again in using a diagram we've seen before, what the inverse problem is. And why it is that Bayes' Theorem is really not very helpful in vision, even though it's widely used as a concept. So remember the situation, you have an illuminant, you have reflectance properties of object surfaces. You have an atmosphere that influences the stimulus that reaches the retina, so the stimulus on the retina you'll recall is an entanglement of all the properties. These are just three of the many properties in the world that go to making up the stimulus. And there's no logical way to get back from the stimulus to the real world features in the world. To know what should the behavior be based on the stimulus when the stimulus just doesn't contain the information that's needed. It contains it in some sense, but you can't get back to that information by any algorithm that reverse engineers the retinal image. So, what does this mean for Bayes' Theorem and its application as a way of thinking conceptually about how vision might be working? Well, the conflation of features in retinal image really precludes biological sensors all over the apparatus of the eye and what follows in the nervous system from apprehending states of the world. This means that B in Bayes' Theorem, let's go back and take a look at Bayes' Theorem, it means that B, these states of the world that are critical In using Bayes' Theorem in any context, is just not available. You don't know what the state of the world is. Why? Because you don't have this information. This information is the state of the world. The visual system can't get it from the retinal stimulus, and so biologically you're just stunned. It's a perfectly good, rational explanation that has value in that regard with respect to kind of putting Helmholtz's original idea that you need inferences of some kind to move forward in understanding how vision works. But Bayes' Theorem, which is the primary way in which the idea is conceptualized today, just can't do the job in the sense that the states of the world are not available because of the inverse problem. That doesn't mean that Bayes' Theorem has no value for vision. It's a rational way of thinking about what's going on, in terms of Helmholzian inference, but it's just like going to work to tell you how vision operates because states of the world can't be known.
