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Abstract. Many applications in science require that computational models and data becombined. In a Bayesian framework, this is usually done by defininglikelihoods based on the mismatch of model outputs and data. However,matching model outputs and data in this way can be unnecessary or impossible.For example, using large amounts of steady state data is unnecessary becausethese data are redundant. It is numerically difficult to assimilate data inchaotic systems. It is often impossible to assimilate data of a complexsystem into a low-dimensional model. As a specific example, consider alow-dimensional stochastic model for the dipole of the Earth's magneticfield, while other field components are ignored in the model. The aboveissues can be addressed by selecting features of the data, and defininglikelihoods based on the features, rather than by the usual mismatch of modeloutput and data. Our goal is to contribute to a fundamental understanding ofsuch a feature-based approach that allows us to assimilate selected aspectsof data into models. We also explain how the feature-based approach can beinterpreted as a method for reducing an effective dimension and derive newnoise models, based on perturbed observations, that lead to computationallyefficient solutions. Numerical implementations of our ideas are illustratedin four examples.