It is a common causal inference problem that, even with theoretically infinite samples, we might be able to only provide bounds for the parameters of interest. This problem occurs naturally, for example, in estimating causal interaction between two risk factors and in estimating the average causal effect using the instrumental variable or Mendelian randomization method. Current procedures include linear programming to get the estimated bounds, plus bootstrapping to get confidence intervals. We describe a likelihood-based procedure that automatically yields the interval estimate from the flat likelihood region and show some theory that allows us to construct confidence intervals from this non-regular likelihood. Finally, we illustrate the procedure with examples from the estimation of causal interaction between two risk factors and the treatment effect under partial compliance.
