Insensitivety to initial condition/prior in data assimilation for the case of the optimal filter and deterministic model

Data assimilation is a term used to describe efforts to improve our knowledge
of a system by combining incomplete observations with imperfect models.
This is more generally known as filtering, which is ’optimal’ estimation of
the state of a system as it evolves over time, in the mean square sense. In
a Bayesian framework, the optimal filter is therefore naturally a sequence of
conditional probabilities of a signal given the observations and can be up-
dated recursively with new observations with Bayes’ formula. When, the
dynamics and observations errors are linear, this is equivalent to the Kalman
filter. In the nonlinear case, deriving an explicit form for the posterior dis-
tribution is in general not possible.
One of the important difficulties with applying the nonlinear filter in practice
is that the initial condition, the prior, is required to initialise the filtering.
However we are unlikely to know the correct initial distribution accurately
or at all. A filter is called stable if it is insensitive with respect to the
prior, that is, it converges to the same distribution, regardless of the initial
condition.
A body of work exists showing stability of the filter which rely on the stochas-
ticity of the underlying dynamics. In contrast, we show stability of the op-
timal filter for a class of nonlinear and deterministic dynamical systems and
our result relies on the intrinsic chaotic properties of the dynamics. We build
on the considerable knowledge that exists on the existence of SRB measures
in uniformly hyperbolic dynamical systems and we view the conditional prob-
abilities as SRB measures ‘conditional on the observation’ which are shown
to be absolutely continuous along the unstable manifold. This is in line with
the result of Bouquet, Carrassi et al [1] regarding data assimilation in the
“unstable subspace”, where they show stability of the filter if the unstable
and neutral subspaces are uniformly observed.

[1] M. Bocquet et al. “Degenerate Kalman Filter Error Covariances and
Their Convergence onto the Unstable Subspace”. In: SIAM/ASA Jour-
nal on Uncertainty Quantification 5.1 (2017), pp. 304–333. url: https:
//doi.org/10.1137/16M1068712.