  When data contains measurement errors, it is necessary to make assumptions
relating the observed, erroneous data to the unobserved true phenomena of
interest. These assumptions should be justifiable on substantive grounds, but
are often motivated by mathematical convenience, for the sake of exactly
identifying the target of inference. We adopt the view that it is preferable to
present bounds under justifiable assumptions than to pursue exact
identification under dubious ones. To that end, we demonstrate how a broad
class of modeling assumptions involving discrete variables, including common
measurement error and conditional independence assumptions, can be expressed as
linear constraints on the parameters of the model. We then use linear
programming techniques to produce sharp bounds for factual and counterfactual
distributions under measurement error in such models. We additionally propose a
procedure for obtaining outer bounds on non-linear models. Our method yields
sharp bounds in a number of important settings -- such as the instrumental
variable scenario with measurement error -- for which no bounds were previously
known.
