Tag: statistics
All the articles with the tag "statistics".
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Sudakov minoration, or how big a maximum must be
Posted on:Sudakov minoration lower-bounds the expected maximum of many Gaussians: if no two are too alike, the maximum is at least of order ε√(log N). The engine behind many impossibility proofs.
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Bias-variance is a Pythagorean decomposition
Posted on:MSE = bias² + variance is the Pythagorean theorem in L²: a constant bias and a mean-zero residual are orthogonal, so their squared lengths add.
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Pólya's recurrence theorem
Posted on:Simple random walk returns to the origin with probability one in one and two dimensions, but not in three. The transition reduces to the convergence of a single series.
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High-dimensional Gaussians live on a sphere
Posted on:The bell-curve picture says Gaussian samples live near the mean. In high dimensions almost all the mass lies in a thin shell at radius √d: density and mass are not the same thing.
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Central Limit Theorem - why sums become Gaussian
Posted on:Adding random variables convolves their densities, and convolution smooths. A geometric look at the central limit theorem, with a slider over the number of summands.
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The 100 prisoners problem
Posted on:100 prisoners each seek their own number in 100 boxes, opening at most 50. Random guessing succeeds once in a nonillion; following cycles succeeds about 31% of the time.
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Optimal message passing on sparse graphs
Posted on:Our NeurIPS 2023 paper: the asymptotically Bayes-optimal classifier for node classification on sparse contextual stochastic block models, and what it implies for graph neural networks.
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Marchenko-Pastur and the Wigner semicircle
Posted on:The eigenvalue histogram of a large random matrix converges to a deterministic shape: Marchenko-Pastur for sample covariance matrices, the Wigner semicircle for symmetric i.i.d. entries.