  The broad concept of emergence is instrumental in various of the most
challenging open scientific questions -- yet, few quantitative theories of what
constitutes emergent phenomena have been proposed. This article introduces a
formal theory of causal emergence in multivariate systems, which studies the
relationship between the dynamics of parts of a system and macroscopic features
of interest. Our theory provides a quantitative definition of downward
causation, and introduces a complementary modality of emergent behaviour --
which we refer to as causal decoupling. Moreover, the theory allows practical
criteria that can be efficiently calculated in large systems, making our
framework applicable in a range of scenarios of practical interest. We
illustrate our findings in a number of case studies, including Conway's Game of
Life, Reynolds' flocking model, and neural activity as measured by
electrocorticography.
