# Gauge Freedom within the Class of Linear Feedback Particle Filters

## TL;DR

This paper explores the gauge freedom in linear feedback particle filters, revealing how different particle transformations can preserve the posterior distribution, thus extending the class of known linear FPFs.

## Contribution

It characterizes gauge freedom in linear FPFs for Gaussian filtering, expanding the set of parametrized linear feedback particle filters.

## Key findings

- Identifies gauge transformations that leave the posterior invariant.
- Extends known families of linear FPFs through gauge characterization.
- Provides a framework for understanding equivalence classes of particle filters.

## Abstract

Feedback particle filters (FPFs) are Monte-Carlo approximations of the solution of the filtering problem in continuous time. The samples or particles evolve according to a feedback control law in order to track the posterior distribution. However, it is known that by itself, the requirement to track the posterior does not lead to a unique algorithm. Given a particle filter, another one can be constructed by applying a time-dependent transformation of the particles that keeps the posterior distribution invariant. Here, we characterize this gauge freedom within the class of FPFs for the linear-Gaussian filtering problem, and thereby extend previously known parametrized families of linear FPFs.