Causal inference with imperfect instrumental variables

Abstract Instrumental variables allow for quantification of cause and effect relationships even in the absence interventions. To achieve this, a number causal assumptions must be met, most important which is independence assumption, states that instrument any confounding factor independent. However, if this condition not can we still work with imperfect instrumental variables? Imperfect instruments manifest themselves by violations inequalities constrain set correlations scenario. In article, establish quantitative relationship between such minimal amount measurement dependence required to explain them case discrete observed variables. As result, provide adapted are valid presence relaxed assumption This allows adaptation existing new lower bounds on average scenarios binary outcomes. Finally, discuss our findings context quantum mechanics.