  In this paper, we develop a kernel learning backward SDE filter method to
estimate the state of a stochastic dynamical system based on its partial noisy
observations. A system of forward backward stochastic differential equations is
used to propagate the state of the target dynamical model, and Bayesian
inference is applied to incorporate the observational information. To
characterize the dynamical model in the entire state space, we introduce a
kernel learning method to learn a continuous global approximation for the
conditional probability density function of the target state by using discrete
approximated density values as training data. Numerical experiments demonstrate
that the kernel learning backward SDE is highly effective and highly efficient.
