  This paper presents a novel distribution-agnostic Wasserstein distance-based
estimation framework. The goal is to determine an optimal map combining prior
estimate with measurement likelihood such that posterior estimation error
optimally reaches the Dirac delta distribution with minimal effort. The
Wasserstein metric is used to quantify the effort of transporting from one
distribution to another. We hypothesize that minimizing the Wasserstein
distance between the posterior error and the Dirac delta distribution results
in optimal information fusion and posterior state uncertainty. Framework
validation is demonstrated by the successful recovery of the classical Kalman
filter for linear systems with Gaussian uncertainties. Notably, the proposed
Wasserstein filter does not rely on particle representation of uncertainty.
Furthermore, the classical result for the Gaussian Sum Filter (GSF) is
retrieved from the Wasserstein framework. This approach analytically exhibits
the suboptimality of GSF and enables the use of nonlinear optimization
techniques to enhance the accuracy of the Gaussian sum estimator.
