# Nonlinear Kalman Filtering with Divergence Minimization

## TL;DR

This paper introduces novel Monte Carlo-based algorithms for nonlinear Kalman filtering that directly optimize divergence measures like KL and alpha-divergence, improving accuracy over traditional filters.

## Contribution

It proposes unbiased algorithms for divergence optimization in nonlinear Kalman filtering without approximations, enhancing filtering performance in complex applications.

## Key findings

- Improved accuracy over UKF and EKF in tracking and pricing tasks.
- Competitive performance with particle filtering.
- Demonstrated effectiveness on radar, sensor tracking, and options pricing.

## Abstract

We consider the nonlinear Kalman filtering problem using Kullback-Leibler (KL) and $\alpha$-divergence measures as optimization criteria. Unlike linear Kalman filters, nonlinear Kalman filters do not have closed form Gaussian posteriors because of a lack of conjugacy due to the nonlinearity in the likelihood. In this paper we propose novel algorithms to optimize the forward and reverse forms of the KL divergence, as well as the alpha-divergence which contains these two as limiting cases. Unlike previous approaches, our algorithms do not make approximations to the divergences being optimized, but use Monte Carlo integration techniques to derive unbiased algorithms for direct optimization. We assess performance on radar and sensor tracking, and options pricing problems, showing general improvement over the UKF and EKF, as well as competitive performance with particle filtering.