The Role of Likelihood in Interval Estimation

It is common practice to treat coverage probability as the only requirement for interval estimation. This allows the use of one-sided confidence intervals. These achieve their coverage probabilities by including parameter values with relative likelihood approaching 0. This violates the law of likelihood, a post-data concept of statistical evidence introduced by Ronald Fisher in 1921. Coverage probability and the law of likelihood are both necessary for interval estimation, but neither is sufficient. When combined, they minimize the possibility of users with the same data, model, and confidence level getting different answers.