A concise mathematical description of active inference in discrete time

Hello, welcome everyone. It is May 30th, 2025. We're in Active Inference Math Stream 14.1. And here with Jesse Van Ustrom discussing a concise mathematical description of active inference in discrete time. This will be a very useful and epistemic discussion. So, Jesse, thank you for writing this paper and coming on to you to introduce it and start walking us through it. Yeah, thank you, Daniel. Thank you for the opportunity of being here and allowing me to present my paper. Yeah, first of all, I wrote this paper together with my colleague, Carlotta Langer and my supervisor, Nia Tai. We are all in Hamburg at the Institute of Data Science Foundations. And yeah, about three years ago, I started getting interested in active inference and wanted to know really exactly how it works, so to say. But this, I had quite a hard time figuring this out and that's a bit why this paper came about. And maybe I can, I also have some notes here. I can, so maybe I can make very specific my question. So what I wanted to know was, okay, we have an agent who has performed a 1 up to a 1 up to a 1 up to a 1 up to a 1. And then what I wanted to know was an an agent acting according to active inference. What will its action 80? How will it choose that? And this question for me was so with the literature at the time, I found it hard to really pin it down. So to say, so to say, so that was the first main objective of making that more clear and especially also making it clear in a way that is with in notation that is very accessible to people who have studied probability theory or any other type of standard mathematics. And yeah. And then throughout the process, a second question came up. So that was the following. So, so I will go into more detail on this later, but agent has something that's called a generative model and how to understand that is as follows. So, uh, the world, we can first look at the real world and that is, uh, generating observations. For example, observation one world is in world state one, then, uh, observation two. So this is all happening in the world, but then an agent. So this is what we could call a generative process. And, uh, uh, uh, generative model is as follows. So this, this is some kind of where an agent has, uh, maybe some kind of simplified version of what's happening in the world. It is representing in its brain. And, um, from that world, from that, some kind of simplified world state, it can predict, okay, what kind of observations can I expect, um, from this state. And this is some kind of a very similar model. And in both cases, we can also actually include actions. So if it acts on the world, it will influence its next world, some kind of simplified world state. And it's the same also here, it will also influence the real world state. So this is called a generative model. And, um, this generative model, it uses it to, um, to choose its next action. But then the second question is, how does it learn, uh, uh, uh, how does it learn this model? So, um, this papers, basically, we can basically divide it into two parts. One part is called inference and the other part. So this is, we could say also action selection and another part is called learning. The generative model. So, um, this presentation will also be, um, somehow divided into these two parts. So I'll first go further into this first question. How does the agent select its actions? And then also, uh, the second later on, I will go on to in the second part, how this is learning the generative model. And, uh, please, if anything is unclear, please feel free to ask questions. If you're now in the live chat, or if you're watching this later, you're also always welcome to send me an email or get in touch in another way. Um, if you have any questions, I'm really happy. So I wrote this paper for people in my situation three years ago, who really want to know the exact mathematical details of how active inference is formulated and, uh, want some kind of an accessible,…