Signal processing makes extensive use of point estimators and accompanying error bounds. These work well up until the likelihood function has two or more high peaks. When it is important for an estimator to remain reliable, it becomes necessary to consider alternatives, such as set estimators. An obvious first choice might be confidence intervals or confidence regions, but there can be difficulties in computing and interpreting them (and sometimes they might still be blind to multiple peaks in the likelihood). Bayesians seize on this to argue for replacing confidence regions with credible regions. Yet Bayesian statistics require a prior, which is not always a natural part of the problem formulation. This paper demonstrates how a re-interpretation of the prior as a weighting function makes an otherwise Bayesian estimator meaningful in the frequentist context. The weighting function interpretation also serves as a reminder that an estimator should always be designed in the context of its intended application; unlike a prior which ostensibly depends on prior knowledge, a weighting function depends on the intended application. This paper uses the time-of-arrival (TOA) problem to illustrate all these points. It also derives a basic theory of region-based estimators distinct from confidence regions.

翻译：信号处理广泛使用点估计量及其伴随的误差界。当似然函数仅存在单峰时，这些方法效果良好，但若似然函数出现两个及以上高峰，则需考虑替代方案（例如集合估计量）以保证估计量的可靠性。置信区间或置信区域或许是直观的首选，但其计算与解释常存在困难（有时仍可能忽略似然函数的多峰特性）。贝叶斯学派借此主张以可信区域替代置信区域。然而贝叶斯统计需要先验分布，这并非总是问题建模的自然组成部分。本文通过将先验重新阐释为加权函数，论证了原本属于贝叶斯框架的估计量如何在频率学派语境中获得意义。加权函数的阐释也提醒我们：估计量的设计应始终基于其预期应用场景；与表面依赖先验知识的先验分布不同，加权函数直接取决于目标应用。本文以到达时间估计问题为例阐释上述观点，并建立了区别于置信区域的基于区域的估计量基础理论。