  Any regular Gaussian probability distribution that can be represented by an
AMP chain graph (CG) can be expressed as a system of linear equations with
correlated errors whose structure depends on the CG. However, the CG represents
the errors implicitly, as no nodes in the CG correspond to the errors. We
propose in this paper to add some deterministic nodes to the CG in order to
represent the errors explicitly. We call the result an EAMP CG. We will show
that, as desired, every AMP CG is Markov equivalent to its corresponding EAMP
CG under marginalization of the error nodes. We will also show that every EAMP
CG under marginalization of the error nodes is Markov equivalent to some LWF CG
under marginalization of the error nodes, and that the latter is Markov
equivalent to some directed and acyclic graph (DAG) under marginalization of
the error nodes and conditioning on some selection nodes. This is important
because it implies that the independence model represented by an AMP CG can be
accounted for by some data generating process that is partially observed and
has selection bias. Finally, we will show that EAMP CGs are closed under
marginalization. This is a desirable feature because it guarantees parsimonious
models under marginalization.
