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Understanding Stein's Paradox (2021)
An intuitive explanation of the James-Stein estimator.
Representing the entire distribution by a single point is not particularly sophisticated, and there is no special reason to choose to use the mode instead of the mean or median except that it happens to give an answer that corrseponds to the James-Stein estimator. But in higher dimensions, the volume subtended is smaller, so that as time progresses you are less and less likely to make your back and the probability approaches zero even in the limit of an infinite number of steps. While rigorously proving this correspondance is non-trivial, both problems have at their core a comparison between the distance between a point and the origin and the volume of the unit sphere in the space.
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