  The main approach to defining equivalence among acyclic directed causal
graphical models is based on the conditional independence relationships in the
distributions that the causal models can generate, in terms of the Markov
equivalence. However, it is known that when cycles are allowed in the causal
structure, conditional independence may not be a suitable notion for
equivalence of two structures, as it does not reflect all the information in
the distribution that is useful for identification of the underlying structure.
In this paper, we present a general, unified notion of equivalence for linear
Gaussian causal directed graphical models, whether they are cyclic or acyclic.
In our proposed definition of equivalence, two structures are equivalent if
they can generate the same set of data distributions. We also propose a weaker
notion of equivalence called quasi-equivalence, which we show is the extent of
identifiability from observational data. We propose analytic as well as
graphical methods for characterizing the equivalence of two structures.
Additionally, we propose a score-based method for learning the structure from
observational data, which successfully deals with both acyclic and cyclic
structures.
