A Concise Mathematical Description of Active Inference in Discrete Time

In this paper we present a concise mathematical description of active inference in discrete time. The main part of the paper serves as a general introduction to the topic, including an example illustrating the theory on action selection. In the appendix the more subtle mathematical details are discussed. This part is aimed at readers who have already studied the active inference literature but struggle to make sense of the mathematical details and derivations. Throughout the whole manuscript, special attention has been paid to adopting notation that is both precise and in line with standard mathematical texts. All equations and derivations are linked to specific equation numbers in other popular text on the topic. Furthermore, Python code is provided that implements the action selection mechanism described in this paper and is compatible with pymdp environments.

On the Fisher-Rao Gradient of the Evidence Lower Bound

This article studies the Fisher-Rao gradient, also referred to as the natural gradient, of the evidence lower bound, the ELBO, which plays a crucial role within the theory of the Variational Autonecoder, the Helmholtz Machine and the Free Energy Principle. The natural gradient of the ELBO is related to the natural gradient of the Kullback-Leibler divergence from a target distribution, the prime objective function of learning. Based on invariance properties of gradients within information geometry, conditions on the underlying model are provided that ensure the equivalence of minimising the prime objective function and the maximisation of the ELBO.

Inversion of Bayesian Networks

Variational autoencoders and Helmholtz machines use a recognition network (encoder) to approximate the posterior distribution of a generative model (decoder). In this paper we study the necessary and sufficient properties of a recognition network so that it can model the true posterior distribution exactly. These results are derived in the general context of probabilistic graphical modelling / Bayesian networks, for which the network represents a set of conditional independence statements. We derive both global conditions, in terms of d-separation, and local conditions for the recognition network to have the desired qualities. It turns out that for the local conditions the property perfectness (for every node, all parents are joined) plays an important role.

Invariance Properties of the Natural Gradient in Overparametrised Systems

The natural gradient field is a vector field that lives on a model equipped with a distinguished Riemannian metric, e.g. the Fisher-Rao metric, and represents the direction of steepest ascent of an objective function on the model with respect to this metric. In practice, one tries to obtain the corresponding direction on the parameter space by multiplying the ordinary gradient by the inverse of the Gram matrix associated with the metric. We refer to this vector on the parameter space as the natural parameter gradient. In this paper we study when the pushforward of the natural parameter gradient is equal to the natural gradient. Furthermore we investigate the invariance properties of the natural parameter gradient. Both questions are addressed in an overparametrised setting.