  Causal inference concerns the identification of cause-effect relationships
between variables, e.g. establishing whether a stimulus affects activity in a
certain brain region. The observed variables themselves often do not constitute
meaningful causal variables, however, and linear combinations need to be
considered. In electroencephalographic studies, for example, one is not
interested in establishing cause-effect relationships between electrode signals
(the observed variables), but rather between cortical signals (the causal
variables) which can be recovered as linear combinations of electrode signals.
  We introduce MERLiN (Mixture Effect Recovery in Linear Networks), a family of
causal inference algorithms that implement a novel means of constructing causal
variables from non-causal variables. We demonstrate through application to EEG
data how the basic MERLiN algorithm can be extended for application to
different (neuroimaging) data modalities. Given an observed linear mixture, the
algorithms can recover a causal variable that is a linear effect of another
given variable. That is, MERLiN allows us to recover a cortical signal that is
affected by activity in a certain brain region, while not being a direct effect
of the stimulus. The Python/Matlab implementation for all presented algorithms
is available on https://github.com/sweichwald/MERLiN
