Tag: statistics
All the articles with the tag "statistics".
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The 100 prisoners problem
Posted on:100 prisoners must each find their own number among 100 randomly filled boxes, opening at most 50 each. Random guessing succeeds with probability one in a nonillion. A particular cycle-following strategy succeeds about 31% of the time. The reason is the cycle structure of a random permutation.
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Optimal message passing on sparse graphs
Posted on:A condensed walkthrough of our NeurIPS 2023 paper deriving the asymptotically Bayes-optimal classifier for node classification on sparse contextual stochastic block models, and what it implies for the design of graph neural networks.
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Marchenko-Pastur and the Wigner semicircle
Posted on:The eigenvalues of a large random matrix do not scatter around. They concentrate, as a histogram, on a deterministic shape. For sample covariance matrices the shape is Marchenko-Pastur; for symmetric matrices with i.i.d. entries it is the Wigner semicircle. Both shapes are computable, and they explain precisely why high-dimensional covariance estimation is biased.
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Stein's paradox
Posted on:In three or more dimensions, the sample mean is dominated everywhere by a shrinkage estimator. The geometric reason is the Gaussian shell: noise pushes you outward, and pulling back is uniformly better. A precursor of ridge regression and most modern regularization.
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Nearest neighbor breaks in high dimensions
Posted on:In high dimensions, all pairwise distances become essentially equal. Nearest and farthest neighbor are no longer meaningfully different. A short geometric tour of the curse of dimensionality.
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St. Petersburg paradox
Posted on:An analysis of the St. Petersburg paradox, where the expected winning value is infinite. Discusses the conflict between mathematical expectation and intuition, and resolves it using practical constraints.