  Data assimilation (DA) provides a general framework for estimation in
dynamical systems based on the concepts of Bayesian inference. This constitutes
a common basis for the different linear and nonlinear filtering and smoothing
techniques which gives a better understanding of the characteristics and
limitations of each approach. In this study, four nonlinear filters for systems
governed by stochastic continuous dynamics with discrete measurements are
inferred as different approximate solutions to the Fokker-Planck equation in
the prediction step of the Bayesian update. The characteristics and drawbacks
of each filtering approach are discussed in light of the introduced
approximations to the original Bayesian problem in each case. The introduced
continuous-discrete (CD) filters are then implemented to solve the combined
estimation problem of the state and parameter variables in a nonlinear system
described by a stochastic differential equation (SDE), namely, the continuous
stirred-tank reactor (CSTR). The performance measures for each filter, such as
estimation errors, computational cost, and the ability to quantify the correct
uncertainty, are compared and investigated with attribution to the assumed
approximations introduced in the derivation of each class of filters.
