  Given a set of several inputs into a system (e.g., independent variables
characterizing stimuli) and a set of several stochastically non-independent
outputs (e.g., random variables describing different aspects of responses), how
can one determine, for each of the outputs, which of the inputs it is
influenced by? The problem has applications ranging from modeling pairwise
comparisons to reconstructing mental processing architectures to conjoint
testing. A necessary and sufficient condition for a given pattern of selective
influences is provided by the Joint Distribution Criterion, according to which
the problem of "what influences what" is equivalent to that of the existence of
a joint distribution for a certain set of random variables. For inputs and
outputs with finite sets of values this criterion translates into a test of
consistency of a certain system of linear equations and inequalities (Linear
Feasibility Test) which can be performed by means of linear programming. The
Joint Distribution Criterion also leads to a metatheoretical principle for
generating a broad class of necessary conditions (tests) for diagrams of
selective influences. Among them is the class of distance-type tests based on
the observation that certain functionals on jointly distributed random
variables satisfy triangle inequality.
