Active Inference in Discrete State Spaces from First Principles

Hello, everyone. Welcome. It's January 15th, 2026, and we're in Active Inference Model Stream number 19.1 with Patrick Kenney and Thomas Parr and Norsegetus guests discussing Active Inference in discrete state spaces from first principles. Patrick has a presentation, and also we will hear from our guests and read any questions from the live chat. So, Patrick, to you for the presentation, and thank you. Go for it. Thanks, Daniel. So, my title is Active Inference in Discrete State Spaces from First Principles. Now, let me explain briefly why discrete state spaces. It's because what I think is a natural formulation of active inference is the path integral formulation, and this is very easy to get to grips with in discrete state spaces because you can do it with a hidden market model as the dynamic model. It's not so obvious what you can do in continuous state spaces with continuous time if you want to do the path integral formulation and develop it to the point where it might actually be of some use in practical applications. And when I say from first principles, that's a nod, if you like, to the free energy principle, I'm proposing that active inference, the problems that need to be addressed can be tackled by standard methods that have been developed in machine learning, in particular the mean field approximation, without having to appeal to the free energy principle. And I'll explain that there are some limitations that this discipline imposes, but also some advantages which accrue from adhering to traditional well understood methods that have been developed in machine learning, Bayesian machine learning, obviously. So, I'll explain then the mean field approximation, how that that addresses the fundamental problem in active inference, the processing the perception action cycle so that an agent can infer its future actions from its past history, essentially by predicting its future sensations. And how this leads to an optimization criterion, in fact, it's a K-elvergence criterion, which is not quite the same as expected free energy. It's very similar, but there is a subtle difference which I will dwell on a bit. And I will explain that there are other natural applications of the mean field of approximation in active inference, which do not appeal to the free energy principle. In particular, the problem of Bayesian learning of HMMs is a direct application of the mean field approximation. Also, the problem of updating beliefs about policies, that too is amenable to treatment by the mean field approximation. So, I'll conclude with the pros and cons of doing things this way. So, here's the mean field approximation in its most compact form. You have a target distribution which you want to approximate, is known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to be known to P. This is the form of the divergence which is capable of being evaluated under these assumptions that Q is tractable and P is intractable. And the basic assumption is very simple. You assume that Q factorizes in this way so that the state space can be split up into streams which are statistically independent under the approximating distribution, not necessarily…