Point Mass in the Confidence Distribution: Is it a Drawback or an Advantage?

Abstract: Stein's (1959) problem highlights the phenomenon called the probability dilution in high dimensional cases, which is known as a fundamental deficiency in probabilistic inference. The satellite conjunction problem also suffers from probability dilution that poor-quality data can lead to a dilution of collision probability. Though various methods have been proposed, such as generalized fiducial distribution and the reference posterior, they could not maintain the coverage probability of confidence intervals (CIs) in both problems. On the other hand, the confidence distribution (CD) has a point mass at zero, which has been interpreted paradoxical. However, we show that this point mass is an advantage rather than a drawback, because it gives a way to maintain the coverage probability of CIs. More recently, `false confidence theorem' was presented as another deficiency in probabilistic inferences, called the false confidence. It was further claimed that the use of consonant belief can mitigate this deficiency. However, we show that the false confidence theorem cannot be applied to the CD in both Stein's and satellite conjunction problems. It is crucial that a confidence feature, not a consonant one, is the key to overcome the deficiencies in probabilistic inferences. Our findings reveal that the CD outperforms the other existing methods, including the consonant belief, in the context of Stein's and satellite conjunction problems. Additionally, we demonstrate the ambiguity of coverage probability in an observed CI from the frequentist CI procedure, and show that the CD provides valuable information regarding this ambiguity.