Tag: statistics
All the articles with the tag "statistics".
-
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.
-
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.
-
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.
-
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.
-
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.
-
Stein's paradox
Posted on:In three or more dimensions the sample mean is dominated everywhere by a shrinkage estimator. The reason is the Gaussian shell, and the idea is a precursor of ridge regression.
-
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.
-
St. Petersburg paradox
Posted on:A coin-toss game whose expected winnings are infinite, and why nobody would pay much to play: expectation against intuition, resolved by practical constraints.