# Smoothing and filtering with a class of outer measures

## TL;DR

This paper introduces a novel filtering and smoothing framework using outer measures to represent uncertainty, generalizing classical Bayesian methods and recovering the Kalman filter under weaker assumptions.

## Contribution

It develops a new approach to filtering and smoothing with outer measures, extending Bayesian equations by replacing integrals with supremums and probability densities with positive functions.

## Key findings

- Framework retains structure of classical Bayesian filtering
- Kalman filter recursion is recovered under weaker assumptions
- Provides a generalized method for uncertainty propagation

## Abstract

Filtering and smoothing with a generalised representation of uncertainty is considered. Here, uncertainty is represented using a class of outer measures. It is shown how this representation of uncertainty can be propagated using outer-measure-type versions of Markov kernels and generalised Bayesian-like update equations. This leads to a system of generalised smoothing and filtering equations where integrals are replaced by supremums and probability density functions are replaced by positive functions with supremum equal to one. Interestingly, these equations retain most of the structure found in the classical Bayesian filtering framework. It is additionally shown that the Kalman filter recursion can be recovered from weaker assumptions on the available information on the corresponding hidden Markov model.