  The reconstruction from observations of high-dimensional chaotic dynamics
such as geophysical flows is hampered by (i) the partial and noisy observations
that can realistically be obtained, (ii) the need to learn from long time
series of data, and (iii) the unstable nature of the dynamics. To achieve such
inference from the observations over long time series, it has been suggested to
combine data assimilation and machine learning in several ways. We show how to
unify these approaches from a Bayesian perspective using
expectation-maximization and coordinate descents. In doing so, the model, the
state trajectory and model error statistics are estimated all together.
Implementations and approximations of these methods are discussed. Finally, we
numerically and successfully test the approach on two relevant low-order
chaotic models with distinct identifiability.
