The problem of data assimilation is to determine the best estimate of the solution history of a dynamical system given some partial and inaccurate measurements. The filtering problem is defined as that of estimating the present state given prior observations. It is generally accepted that the optimal solution is obtained by calculating the conditional statistical distribution of the state vector of the system given the measurements up to the current time.
The conditional probability density function (PDF) solves the forward Kolmogorov equation between measurements, and the PDF is updated by Bayes's rule at measurement times. However, sovling the forward Klmogorov equation for multidimensional, many -variable systems is not a practical. One efficient way to circumvent this difficulty is to evolve the statistics by computing an N-sample ensemble of realizations of the system, for example, Sequential Importance Resampling method, Ensemble Kalman Filter, and Maximum Entropy Filter. I'll compare those methods and present results for a simple stochastic PDE model of the ocean thermohaline circulation, which has bimodal statistics associated to two distinct stable states.