Foundations of Probability & Inference

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Abstractions about the basic apparatus of probability theory and statistical inference — foundational objects (random variables, probability distributions, empirical measures), limit theorems and inequalities (Lévy's continuity theorem, Kolmogorov's three-series theorem, Jensen's inequality), information measures and distances (entropy, maximum entropy, Hellinger distance), and pitfalls like the gambler's fallacy.

29 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Blackwell–Girshick Equation — A two-term identity separating the variance of an independent-count random sum into mark-size and count-uncertainty contributions.
• Brownian Skorokhod Embedding — Represent a prescribed probability law as Brownian motion observed at an adapted stopping time, with admissibility conditions stated separately.
• Correlation Dimension — A fractal-dimension statistic given by the small-radius scaling exponent of the probability or normalized count that two sampled points lie within distance epsilon.
• Cramér's Theorem (Large Deviations) — An i.i.d.-sample-mean large-deviation principle whose exponential rate is the convex conjugate of the one-observation log moment-generating function.
• Credal Set — Represent imprecise probabilistic belief by a set of admissible probability measures, deriving lower and upper expectations as envelopes while keeping convexity, closure, conditioning, and independence choices explicit.
• Empirical Measure — The random atomic probability measure P_n = n^{-1} sum_i delta_Xi that assigns equal mass to realized observations and turns sample averages into integration against a measure.
• Exponentially Modified Gaussian Distribution — The distribution of an independent Gaussian value plus a positive exponential value, yielding a precise right-skewed convolution family.
• Gambler's Fallacy — Catch the error of believing a run of one outcome makes the opposite 'due' on the next trial — imposing the law of large numbers' aggregate balance as a within-trial obligation — by first screening whether the trial process is independent.
• Hellinger Distance — A metric between probability laws given, in the normalized convention, by one over square root two times the L2 distance between their square-root densities.
• Independent and Identically Distributed Random Variables — Model a collection of random variables as mutually independent draws from one common probability distribution, separating repeated sampling from dependence and distributional drift.
• Information Entropy — The probability-weighted average of logarithmic surprise across the outcomes of a discrete random variable.
• Jensen's Inequality — For a convex function, the function of a mean is no greater than the mean of the function's values, under the required domain and expectation conditions.
• Kelly's Lemma — Certify a continuous-time Markov chain's stationary law by matching each forward transition's weighted flow to a candidate reverse transition and matching their statewise exit rates.
• Kolmogorov's Three-Series Theorem — An if-and-only-if test for almost-sure convergence of an independent random series using large-jump probabilities, truncated means, and truncated variances.
• Lorden's Inequality — Lorden's inequality bounds the mean excess at first crossing of any nonnegative threshold by a positive-part second moment divided by the positive drift of iid increments.
• Lévy's continuity theorem — Lévy's continuity theorem equates convergence in distribution of probability measures with pointwise convergence of their characteristic functions, subject to continuity of the limiting function at zero.
• Minlos's theorem — Minlos's theorem turns a cylindrical measure on the dual of a nuclear space into a Radon measure when its Fourier transform is continuous.
• Monotone Likelihood Ratio Property — Order a parametric family so every higher-parameter to lower-parameter likelihood ratio is nondecreasing in one statistic, making larger statistics monotonically stronger evidence for the higher parameter.
• Natural Exponential Family — A full one-parameter family of probability laws formed by exponentially tilting one fixed measure by the observed value and normalizing on its finite natural domain.
• Principle of Maximum Entropy — A constrained probability-assignment rule that selects the feasible distribution with greatest entropy relative to a declared reference.
• Probability Distribution — The complete specification of how probability mass or density is spread over a random variable's possible values — a measure that, once compressed to a named parametric family, encodes shape, moments, tails, and a generative claim about the process producing the data.
• Random Variable — Translate an uncertain event into a measurable function X: Ω → ℝ on a probability space, so that a number attaches to each outcome and the whole apparatus of expectation, distribution, and convergence becomes computable before any value is observed.
• Residence Time (Statistics) — The statistical residence time is the expected first exit of a random process from a specified domain, conditional on its starting state.
• Scoring Rule — Evaluate a probabilistic forecast after its outcome by mapping the report–outcome pair to a numeric loss or reward, with propriety governing whether truthful distributions are optimal in expectation.
• Spitzer's Formula — Recover an i.i.d. random walk's running-maximum laws from positive-part partial-sum laws through Spitzer's time-generating-function identity.
• Stochastic Equicontinuity — Stochastic equicontinuity makes large local oscillations of indexed random functions unlikely as their arguments become close.
• Truncated Distribution — A probability law conditioned on a positive-probability retained region and renormalized over that region.
• Tsallis Distribution Family — Organize probability laws obtained from declared Tsallis-entropy constraints around a q-exponential kernel whose support, tails, moments, and classical limit depend on the deformation index and parameterization.
• Yule–Simon Distribution — A one-parameter distribution on positive integers with beta-function probability mass and a power-law tail, associated with cumulative-advantage frequency models.