Textbook Group meeting for Parr, Pezzulo, Friston 2022 .

Hello, it is Cohort 2, March 8th, 2023. We're in meeting 19, and we're in our second discussion on chapter 8. We will look over the questions on chapter 8, and talk about any sections of chapter 8. And we will also probably take a preview look towards 9. So, just at the outset, does anyone have any comments or questions on 8? Anything else that they've come across or learnt over the last week that puts continuous time models in a different context? Any other thoughts on continuous and discrete? Or we'll look over the questions and make sure that we've at least touched upon each of the questions from the previous cohorts. Yeah, I'll let you go for it. Actually, based on some of the claims, especially in a Bayesian mechanics paper, and also in this recent Path Integrals paper, they claim that FEP formulation, at least from about 2012 2012 to 2019, was largely focused on developing density over states mathematics, or in other words, the first two types of FEP path tracking and path matching and path mode dynamics. But, I'm sorry, from that from that from that from that from that from that from that from that from that from that they've concentrated more in developing the path integral formulation. I'm not sure which direction this textbook leans toward, because in some places it seems like it is the extension of the previous developmental trajectories. But especially in Chapter 8, it gets more into this recent formulation of formulating the path integral mathematics. But then again, even in Chapter 8, we see discussions about the... Disappeared. Discuss about the... So, yes, I mean, previous years and this year, and also, yeah, how the textbook leans maps into that dichotomy. Yeah, thank you. Good points, Thomas, overviewed as such. And... This... PAR textbook seems to be more heavily on the mode matching and mode tracking. However, the continuous time... It's kind of like the more things change, the more they stay the same. So, looking at Thomas's overview of the history, the early implementations, as he noted, were in continuous time. And then, in order to bring on explicit planning, discrete cognitive decision-making, symbolic state spaces, and so on, sequential behavior, they actually pivoted to the discrete time. And that left the continuous time lock of Volterra, generalized dynamical systems, phrasings behind. And then, now we have a fusion of the continuous and the discrete times, typified by the figure 8.6, where, through hierarchical nested models, we can include features of discrete decision-making, as well as continuous perception and action. Also, discussed in Livestream 46, ACNIMF does not contradict folk psychology with Kiefer, Ramsted, and Smith, which discuss, essentially, this model as the folk psychological scaffold. That there's a motor active inference layer that's perceptual and active in continuous space and time. And then, we can propose these symbolic, discrete, explicit planning-oriented cognitive models, as cognitive models, I guess. So, it's funny. So, that's one return from the earliest, early 2000s continuous time. Then, the focused move to discrete time. And then, there's the hybrid models. And then, as you pointed to, it even goes deeper, because the discrete time formalism is perfectly suited for the first two phases of Bayesian mechanics. And the non-equilibrium steady state and the state-space representations. Whereas, for the truly path-tracking Bayesian mechanics, and probably G-theory, chaos, and all of this, continuous time is like your starter position before it even goes further afield into gauge and fibers and sheaths and all of these other formalisms. So, it's almost like there's a sequence, because the discrete time and the matrix multiplication are very straightforward. Then, when we look at figure 4.3 and figure 8.6, we kind of see some similarities between continuous and discrete time. And then, from the continuous time, we can take another jump into the gauge setting, which does…