  The principle of the common cause claims that if an improbable coincidence
has occurred, there must exist a common cause. This is generally taken to mean
that positive correlations between non-causally related events should disappear
when conditioning on the action of some underlying common cause. The extended
interpretation of the principle, by contrast, urges that common causes should
be called for in order to explain positive deviations between the estimated
correlation of two events and the expected value of their correlation. The aim
of this paper is to provide the extended reading of the principle with a
general probabilistic model, capturing the simultaneous action of a system of
multiple common causes. To this end, two distinct models are elaborated, and
the necessary and sufficient conditions for their existence are determined.
