  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.
