Abstract

Transfer learning can improve precision by borrowing information from external sources, but how much improvement is possible while maintaining valid inference? We address this by studying honest confidence intervals for a target parameter using multiple sources with heterogeneous prespecified discrepancy bounds. Under local information and admissibility conditions, we establish a universal lower bound on interval length for procedures maintaining uniform coverage. A scalar information equation characterizes the bound, revealing how precision depends jointly on source sample sizes and discrepancies. Large source samples need not yield greater precision gains when their discrepancies are large. When discrepancies are unrestricted, honest intervals cannot improve upon the target-only rate, even when all sources are unbiased. For the Gaussian location model, we calculate optimal source weights and construct honest intervals attaining the lower bound order, establishing sharpness. We extend the lower bound theory to semiparametric models through parametric submodels. For the Cox proportional hazards model, we verify the information conditions under which the lower bound applies, including under right censoring.