Textbook Group meeting for Parr, Pezzulo, Friston 2022 .

okay it's September 16th 24 and Andrew is going to give an overview of chapter four then we'll see what questions we have slash other questions already written so go for it Andre thanks Daniel uh yeah so again this is chapter four active infer uh 2022 textbook uh we're working with cohort 7 so this is the first week of going through chapter 4 so I figured I'd give uh an overview of the chapter uh we're already told off the bat in uh section 4.1 the introduction uh that we're going to be introduced to the relationship here between free energy and basian inference uh the form of typical active inference generative models that we often see in active inference implementations uh and we'll also look at the Dynamics obtained from minimizing these models free energy so specifically we look here at both discreet and continuous time models as well as the neurobiologically motivated idea of inferential message passing underlying the computations and predictive coding involved in active inference um so again this is this is the first half of the textbook so this is much more more Theory rather than direct implementation however chapter 4 is particularly uh mathematically involved uh in comparison to the prior three chapters um of course bases rule figures very strongly here so we see that in section 4.2 from basian inference to free energy um so we recall that we're working with approximate inference rather than exact basian optimality uh given the problem of tractability or intractability uh as well as neurobiological grounding of uh of limited resources so the idea being that for biologically plausible models we have to actually consider uh any kind of um computational limitation of resources or metabolic and other biological uh limitations on the processes that that proceed during pre- energy minimization so that's the kind of matching between biological plausibility and why we're looking at approximate rather than exact uh uh free energy minimization so the uh or rather surprise minimization the mathematics involved here are drawn from linear algebra uh differentiation and calculus and the tailor series expansion uh the textbooks appendices give further details on the relationships between these fields so I strongly REM uh recommend looking at those uh if you if you are still kind of learning the maths involved uh we're shown in equation 4.1 bases rule uh so here this this helps us to better understand a generative model and active inference uh which is a generative model can be described as uh here we can see it as like the probability of x times the probability of Y conditioned on X and then we can flip those around on the other side of the equal sign uh and so that that gives us a way of understanding like we can we can use B's rule to potentially update any aspect or any component of B's role uh whenever something new is updated just kind of uh sort of plug and chug for the rest so to speak right so that means whenever we have like new observations uh come in so our model our our our active inference agent observes a new observation we can update its uh posterior beliefs about States so um so part of how we're able to do this also this is kind of a key aspect to understanding what free energy is relative to surprise and a lot of the other theoretical terms that were introduced to um Jensen's inequality renders optimization here possible since minimizing surprise itself would be next to Impossible it's intractable whenever it comes to like very complex models uh integrals that that Things become rather difficult to keep track of or there are limited computational resources whenever doing the kind of computations involved with numerous or highly complex matrices um we we can use Jensen's inequality which is uh this idea that the log of an average is always greater than or equal to the average of a log um it's you know I don't want to call it a trick but in a in a sense it's what allows us to say oh this uh this tractable uh…