Tag: statistics
All the articles with the tag "statistics".
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Percolation
Posted on:Keep each edge of a grid with probability p and ask whether a path crosses the grid. The answer flips at a sharp threshold, exactly 1/2 on the square lattice by self-duality.
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Shattering and the VC dimension
Posted on:The VC dimension of a class of yes/no rules is the largest set of points it can label in every way. Past it, the labelings stop doubling, and that single integer governs learnability.
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The Hanson–Wright inequality
Posted on:A quadratic form in independent random variables concentrates around the trace: a Gaussian tail near the mean via the Frobenius norm, an exponential tail further out via the operator norm.
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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.