  Data assimilation is the task of combining mathematical models with
observational data. From a mathematical perspective data assimilation leads to
Bayesian inference problems which can be formulated in terms of Feynman-Kac
formulae. In this paper we focus on the sequential nature of many data
assimilation problems and their numerical implementation in form of Monte Carlo
methods. We demonstrate how sequential data assimilation can be interpreted as
time-dependent Markov processes, which is often referred to as the McKean
approach to Feynman-Kac formulae. It is shown that the McKean approach has very
natural links to coupling of random variables and optimal transportation. This
link allows one to propose novel sequential Monte Carlo methods/particle
filters. In combination with localization these novel algorithms have the
potential of beating the curse of dimensionality, which has prevented particle
filters from being applied to spatially extended systems.
