# Approximate Bayesian inference as a gauge theory

## TL;DR

This paper introduces a novel algorithm using Schild's ladder for parallel transport on statistical manifolds, advancing the mathematical framework of gauge theories applied to approximate Bayesian inference in biological systems.

## Contribution

It presents a new algorithm leveraging Schild's ladder for parallel transport in variational inference on manifolds, enhancing the gauge theory approach to neuronal dynamics.

## Key findings

- Provides a formal framework for Bayesian inference as a gauge theory
- Develops an algorithm for parallel transport of statistics on manifolds
- Offers insights into neural phenomena like attention and perception

## Abstract

In a published paper [Sengupta, 2016], we have proposed that the brain (and other self-organized biological and artificial systems) can be characterized via the mathematical apparatus of a gauge theory. The picture that emerges from this approach suggests that any biological system (from a neuron to an organism) can be cast as resolving uncertainty about its external milieu, either by changing its internal states or its relationship to the environment. Using formal arguments, we have shown that a gauge theory for neuronal dynamics -- based on approximate Bayesian inference -- has the potential to shed new light on phenomena that have thus far eluded a formal description, such as attention and the link between action and perception. Here, we describe the technical apparatus that enables such a variational inference on manifolds. Particularly, the novel contribution of this paper is an algorithm that utlizes a Schild's ladder for parallel transport of sufficient statistics (means, covariances, etc.) on a statistical manifold.