  The paper deals with conditional linear information inequalities valid for
entropy functions induced by discrete random variables. Specifically, the
so-called conditional Ingleton inequalities are in the center of interest:
these are valid under conditional independence assumptions on the inducing
random variables. We discuss five inequalities of this particular type, four of
which has appeared earlier in the literature. Besides the proof of the new
fifth inequality, simpler proofs of (some of) former inequalities are
presented. These five information inequalities are used to characterize all
conditional independence structures induced by four discrete random variables.
