Textbook Group meeting for Parr, Pezzulo, Friston 2022 .

All right, greetings. Thanks for joining everyone. It's March 8th, 2023, meeting seven for cohort three. We're in our second discussion on chapter three in the textbook. And last time we went over the whole chapter, kind of ran through it, but many things that we can return to and many questions that people have on chapter three. So to begin with, does anyone, whether they wrote the question previously or not, do they want to just like share something from their reading about three or some insight or question about three or whatever else you want to share right now? Jonathan, and then anyone else? Yeah, I think I'm particularly interested in the question there about about the graphic with the distributions. I don't know if Francois is around at the moment, but I was talking with him about this, this question. There are some, some things we're a bit confused about. So that would be very interesting for me to go through. This one, the pink one. Yeah. All right. Thank you. Any other just general comments? Otherwise we can just start with this 3.2. All right. All right. So figure 3.2. So for the top graphs, these are points sampled from distributions mu or x given b for different values of b blanket states on the x-axis. This is fine. For the bottom left graph, what is meant by the average internal state? The average value of mu for all the points in the top right graph, which are just sampled from conditional distributions. So to get exact value, we would have to take infinite samples or integrate. My question is, how exactly do we, from the bottom left plot, identify the average x and variance around it? And then for the bottom right plot, what exactly is the expected value that x is conditioned upon? The subscript is p of mu given b. Do we select a specific b or double integrate over b as well? All right. Great question. Does anyone want to give a first thought on how anyone can just describe the graph as they see it? Or give a thought on how from the bottom left graph, the bottom right is generated? Jonathan? Jonathan? Yes. Maybe I can talk through what I think and what the confusion for me was. And so if we go to the top graphs there, we're clearly looking there. So x here is sort of, for instance, an external temperature and b is the sensory state. And so it says, how is the external temperature conditioned on the sensory state, even though there's not a causal relationship in that direction? And then how is the internal state determined by the sensory state, by the blanket state? And so I think these two make sense. And then indeed the bottom left, I think also makes some sense in that you can take some particular b and you can say, what's the distribution in both mu and x over b? And that will give you this graph or any particular value of b. I think for me, the confusion here lies in, so the bottom right plot there is written as the expectation where the distribution is over, it's mu given b. And so it's not quite clear there what mu given b would normally, I would normally think of that as you give it a particular value of b. And from that, it's an expectation as the distribution over the mu's and the expectation thereby over the mu's. But it's not clear here that this is actually for a given b. So the question I think is, is that expectation at the bottom right really over mu given b or is it just over mu? Yeah, great question. I'll try to give a first thought, but if anyone sees any other mathematical way, go for it. I believe it is for a specific b. So we're looking at this crosshairs where we'll just continue with this temperature, but we're dealing with an agent whose mapping might be like a little bit wacky with temperature. So when mu is 50, that is corresponding to a blanket reading of negative 5. So if all we got was a b reading of negative 5, our maximum likelihood estimate for mu would be 50. Now, so that's, we're conditioning on a particular observation of b. Now, conditioning on mu being 50, we have a mean…