# The Hitchhiker's Guide to Nonlinear Filtering

## TL;DR

This paper provides a comprehensive tutorial on nonlinear filtering, covering static Bayesian inference, discrete-time models, and continuous-time theory, with insights into probability measures and particle filtering algorithms.

## Contribution

It offers an accessible overview connecting various aspects of nonlinear filtering, serving as an introduction for scientists and engineers to advanced literature.

## Key findings

- Unified view of discrete and continuous filtering
- Insights into probability measure changes in filtering
- Guidance on particle filtering algorithm construction

## Abstract

Nonlinear filtering is the problem of online estimation of a dynamic hidden variable from incoming data and has vast applications in different fields, ranging from engineering, machine learning, economic science and natural sciences. We start our review of the theory on nonlinear filtering from the simplest `filtering' task we can think of, namely static Bayesian inference. From there we continue our journey through discrete-time models, which is usually encountered in machine learning, and generalize to and further emphasize continuous-time filtering theory. The idea of changing the probability measure connects and elucidates several aspects of the theory, such as the parallels between the discrete- and continuous-time problems and between different observation models. Furthermore, it gives insight into the construction of particle filtering algorithms. This tutorial is targeted at scientists and engineers and should serve as an introduction to the main ideas of nonlinear filtering, and as a segway to more advanced and specialized literature.