  Instrumental variables have proven useful, in particular within the social
sciences and economics, for making inference about the causal effect of a
random variable, B, on another random variable, C, in the presence of
unobserved confounders. In the case where relationships are linear, causal
effects can be identified exactly from studying the regression of C on A and
the regression of B on A, where A is the instrument. In the more general case,
bounds have been developed in the literature for the causal effect of B on C,
given observational data on the joint distribution of C, B and A. Using an
approach based on the analysis of convex polytopes, we develop bounds for the
same causal effect when given data on (C,A) and (B,A) only. The bounds
developed are thus in direct analogy to the standard use of instruments in
econometrics, but we make no assumption of linearity. Use of the bounds is
illustrated for experiments with partial compliance. The bounds are, for
example, relevant in genetic epidemiology, where the 'Mendelian instrument' S
represents a genotype, and where joint data on all of C, B and A may rarely be
available but studies involving pairs of these may be abundant. Other examples
of bounding causal effects are considered to show that the method applies to
DAGs in general, subject to certain conditions.
