Abstract: This chapter provides a broad introduction to Bayesian data assimilation that will be useful to practitioners in interpreting algorithms and results, and for theoretical studies developing novel schemes with an understanding of the rich history of geophysical data assimilation and its current directions. The simple case of data assimilation in a ‘perfect’ model is primarily discussed for pedagogical purposes. Some mathematical results are derived at a high level in order to illustrate key ideas about different estimators. However, the focus of this chapter is on the intuition behind these methods, where more formal and detailed treatments of the data assimilation problem can be found in the various references. In surveying a variety of widely used data assimilation schemes, the key message of this chapter is how the Bayesian analysis provides a consistent framework for the estimation problem and how this allows one to formulate its solution in a variety of ways to exploit the operational challenges in the geosciences.

Machine learning (ML) is a data-driven modeling approach that has become popular in recent years, thanks to major advances in software and hardware. Given enough data about a complex system, ML allows a computer model to imitate that system and predict its behavior. Unlike a deductive modeling approach, which requires some understanding of a system to be able to predict its behavior, the inductive approach of ML can predict the behavior of a system without ever understanding it in a traditional sense. Climate is a complex system, but there is not enough observed data describing an unprecedented event like global warming on which a computer model can be trained. Instead, it may be more fruitful to use ML to imitate a climate model, or a component of it, to greatly speed up computations. This will allow the parameter space of climate models to be explored more efficiently.