Textbook Group meeting for Parr, Pezzulo, Friston 2022 .

okay welcome back cohort 6 we're in our first discussion of chapter 4 so Andrew however you'd like to begin and for everyone else any questions in the chat or raise your hand or however and let's jump into four perfect um yeah welcome everyone we're going going over chapter 4 for the first time with cohort 6 um for warning uh for anyone new to active inference this chapter is a bit more mathematically involved than the preceding chapters um but this is where we actually start to relate a lot of the concepts and kind of uh overview ideas that we've already been introduced to we begin to kind of map these uh more mathematically to understand the computations involved inactive inference models um so more specifically this chapter goes through the typical forms of active inference uh generative models will'll see both in discret and continuous time um the relationship between those models and uh the Dynamics of minimizing their free energy um at the very end we're also briefly introduced to other things that we'll see in later chapters including inferential message passing so um I would like to I I think this chapter does well at highlighting the importance of approximate inference um we've seen before that the a lot of the underlying assumptions in active inference uh involves uh the minimization of free energy now really the idea is that agents would want to quote unquote want to uh minimize surprise but because uh we need to actually render the computation tractable we instead are minimizing a quantity known as free energy um quick note that the mathematics involved here draw from different mathematical Fields primarily linear algebra differentiation in in multivariate calculus and the use of tailor series expand and some others um there's some nice appendices in the back of the textbook for anyone who's either new to those uh areas of maths or need some kind of refresher and would like to get a closer look on their relationships specifically with the computations involved here um and so sort of the core here is that we're introduced with a CA equation 4.1 we already know from previous chapters a generative model in a in a certain way can be reduced to a prior and a likelihood um those fit right into basis theorem that's equation 4.1 and so it relates the prior and the likelihood on the left hand side which you could just replace with you know uh oftentimes there's a notation the letter M to denote a model and then on the right side um those end up equaling a posterior distribution which is your your hidden States conditioned on your observations times um why or your your observations so your your evidence and so you can imagine whenever the agent basically takes in new observations all these quantities can be updated um and and that's the kind of nice easier kind of introductory view of how how this works um as far as rendering uh everything track [Music] um we also exploit something called Jensen's inequality um put in simple terms Jensen's inequality says the log of an average is always greater than or equal to the average of a log um so we we can exploit that and rewrite basis theorem using logs um and we use a different notation rather than a capital P that might be used to denote an exact distribution we can use a q which is used to denote an approximate distribution um and and replace that in there and so the whole idea is by minimizing free energy as a kind of proxy that is always greater than or equal to surprise the agent again quote unquote once to minimize free energy in order to minimize its surprise if you can get those two Quant ities to be virtually zero and virtually uh equal to one another that that would be kind of the general overall goal um and to kind of Zoom through a bit more of the rest of what happens in this chapter we get a further breakdown of what comprises free energy um it involves um using Kack uh colak leer Divergence between that approximate distribution the Q I mentioned earlier and uh an…