Confidence intervals for the normal mean utilizing prior information

Consider X1,X2,…,XnX1,X2,…,Xn that are independent and identically N(μ,σ2)N(μ,σ2) distributed. Suppose that we have uncertain prior information that μ=0μ=0. We answer the question: To what extent can a frequentist 1−α1−α confidence interval for μμ utilize this prior information?