Textbook Group meeting for Parr, Pezzulo, Friston 2022 .

okay welcome back cohort 7 we're discussing chapter 4 so anyone can bring up any comment or anything they remembered or a question from four or some other comment from earlier on uh or we'll look at the chapter or look at questions but first does anyone have a comment or just any other anything else they want to kick it off with okay type in the chat or or raise your hand or anything um chapter 4 has two roads leading into it with chapter 2 and three so this is really where we see active inference generative models after the more General context of one and then the low and the high road in in chapters 2 and three um anyone like is there a part of four that anyone wants to go to or just a quote that's interesting just to begin okay 4.1 single paragraph describes in very concise kind of llm like style what the next sections are going to be uh section 4.2 goes from base equation base theorem exact base which is exactly how you update the prior into the posterior given an observation that's a calculation that is done like in one Fell Swoop the idea of using some bound on exact basing inference so the idea of doing approximate basing inference using some other proxy not surprise exactly but abound on surprise like variational free energy the idea here is that that bound could be picked to have certain properties like be able to optimize it incrementally with gradient descent like in machine learning like the title of the section they do start with base theorem though this chapter also says that it will present the active inference generative models and it does there are a few kind of background detours or or um interludes like Jensen's inequality towards equation 4.2 uh they use the definition of prize as log likelihood and then through the usage of that log or just any function that has like a decreasing uh rate of decrease of the slope but never turning back negative um the log surprisal allows like a um strict bound to be described which is called the free energy or negative free energy like just depending on the negative sign um okay Jonathan a previous uh participant and a math teacher I think Professor has written this document let's just see it for a second some of these it's like okay I think this date is automatically chosen but that would be interesting if he updated it so recently let's look through this while there are some superb explanations in the book there are important details left out in the derivations which may not be that important on the first pass but I think make things much Clear when you want to understand it in detail our general end will be to observe things we expect to observe we will come on later to the things that we want to observe erve when we think about preferences but for now we can just think about the things that a model predicts that we are likely to see here's the part with Jensen's inequality so exact base 2.1 just like in the book Jensen's inequality this is going into more lines of detail from what is in the book this gives some um lines to follow this bottom line here here's surprisal log likelihood this is the likelihood of that observation and then log likelihood for for a variety of reasons to to get some 01 uh interval properties to get that boundedness um to be able to compare multiple uh like uh orders of magnitude of probability so this is whatever it is here's the K Divergence between like the distribution that you can control your surprisal from and uh the empirical distribution P um chapter 2.5 or chapter 2 equation 2.5 had the variational free energy and there was multiple lines it seems like he's going into more detail here connecting how free energy described in Chapter 2 and and then expected free energy in equation 2.6 and then connecting it more to chapter 4 so that's that's also kind of a background math like a standalone math thing any ideas or questions on 4.1 or 4.2 so 4.1 again single paragraph yeah Jeff go for it yeah no um I was recently reading a paper and…