Bayesian Model-Based Cognition: The Requirement Equation

Hello and welcome. It is April 29th, 2024. We're in Active Guest Stream 82.1 with Robert Worden discussing Bayesian Model-Based Cognition, the requirement equation. Thank you, Robert, to you for introduction and jumping into a presentation and discussion. Thanks for joining. Hi, I'm Robert Worden and I work Theoretical Neurobiology Group at UCL London. I'm going to talk in this live stream about an equation called the requirement equation, which has a lot to do with free energy principle and active inference, but it's not the same thing. And I hope this is the first of a series of live streams talking about how we apply these ideas to 3D spatial cognition, particularly, and eventually to consciousness. But to get on with this presentation, here's a summary of the key ideas. The main idea of this requirement equation is to ask, not ask what brains have to do, not ask how they do it. And so this approach is different from the free energy principle and active inference, but I believe it complements it in a way I'll try to explain. So, as a statement of what brains have to do, there is this requirement equation, which is a mathematical equation, essentially brains need to compute it or to give the same answer as if they computed it to give the greatest possible fitness to their owners. So that's, it's a statement of what brains need to do. And it's not a statement of how they need to do it. Doesn't say anything about neurons or pre-energy minimization or anything like that. It simply says to get as fit as possible, a brain has to do this. And I'll describe what this is. And it turns out this requirement equation is like Bayes' theorem. It's very similar to Bayes' theorem, but it has some extra terms in it. And you'll see that. And because it's the fittest possible brain, brains have to evolve towards the requirement equation, towards doing that thing. And the trouble is the requirement equation itself is too expensive to compute and brains don't compute it. So animals don't compute it, but they have to produce very similar results to that. But we can compute the requirement equation. We can compute it by brute force. And this makes no assumptions about how brains do it. And so that gives the best possible fitness. That's a kind of yardstick against which you can calibrate any model of cognition. And particularly you can use it to calibrate and test free FEP and active inference models. And I'll put the viewpoint that they are really approximations to this equation. And so we can test active inference models by comparing their results with the requirement equation results. So the first part of this talk is to derive this equation and show you how it works. The second part of the talk is to show how it can be used to test and calibrate active inference models. And the third part in point six is just a few remarks about internal Bayesian models. Okay, so to start, how do we characterize what a brain does without making assumptions about how it does it? And the first characterization is a black box that takes sense data as it inputs and produces choices of actions as it outputs. So that's all a brain does. It takes a set of sense data, D, which can include vision, touch, all sorts of stuff. And then it chooses an action. Now, we want to go a little beyond that black box model. And we're going to what I call a gray box model shown in the lower diagram. And what happens is this. The brain examines it has a set of possible actions, AI. I is one, two, three, etc. Now, for each of those possible actions, it calculates some real function, F of D and A. And that real function depends on both the sense data and the action it's going to choose. So F is a kind of expected payoff for that action in these circumstances. And then the brain compares all the possible Fs. So here we are on the bottom left diagram. It's computing a bunch of different Fs in parallel. And then it puts them all to a comparison. And it picks the biggest one. It picks the…