Tag: statistics
All the articles with the tag "statistics".
-
Percolation
Posted on:Keep each edge of a grid independently with probability p and ask whether a connected path crosses the whole grid. The answer flips from almost-never to almost-always at a sharp critical value, which on the square lattice is exactly 1/2 by a self-duality argument. An interactive grid with a monotone coupling on the slider, the dual blocking wall, and the theorems of Harris, Kesten, and Smirnov.
-
Shattering and the VC dimension
Posted on:How expressive is a family of yes/no rules? Shattering measures it on a fixed set of points: the class can cut them apart in every conceivable way. The VC dimension is the largest set it can shatter, a single integer. Past that size the number of labelings stops doubling and is trapped under a polynomial, the Sauer-Shelah jump, and that one integer is what makes an infinite class learnable.
-
The Hanson–Wright inequality
Posted on:A quadratic form in independent random variables concentrates around its mean, the trace of the matrix. Hanson–Wright pins down the two-sided tail: a Gaussian regime near the mean set by the Frobenius norm, crossing over to a heavier exponential tail set by the operator norm. For Gaussian inputs it all follows from rotating to the eigenbasis and applying Bernstein.
-
Sudakov minoration, or how big a maximum must be
Posted on:Averages shrink, but maxima grow. Sudakov's minoration inequality is the clean tool for the harder direction: a lower bound on the expected maximum of many Gaussians. As long as no two of them are too alike, that maximum is at least ε times the square root of log N. This is the engine behind a lot of impossibility proofs.
-
Bias-variance is a Pythagorean decomposition
Posted on:The textbook derivation of MSE = bias² + variance reads as a cancellation in algebra. Geometrically, it is the Pythagorean theorem in L². A constant bias and a mean-zero residual are orthogonal, and their squared lengths add. The bias-variance tradeoff lives on a right triangle.
-
Pólya's recurrence theorem
Posted on:Simple random walk on the integer lattice returns to the origin with probability one in 1D and 2D. In 3D and higher, there is a positive probability of never returning. The transition is exact, dimension-dependent, and reduces to convergence of a single harmonic-style series.
-
High-dimensional Gaussians live on a sphere
Posted on:The bell-curve picture says Gaussian samples live near the mean. In high dimensions that picture is catastrophically wrong: almost all the mass lies in a thin spherical shell at radius √d. Density and mass are not the same thing.
-
Central Limit Theorem - why sums become Gaussian
Posted on:A geometric look at the central limit theorem. Adding random variables is convolving their densities. Convolution smooths. Watch a Bernoulli, a die roll, or a bimodal distribution become Gaussian as you slide the number of summands.