Physics data analysis — the application of probability, statistics, and inference to experimental physics data. Distinguished from general machine-learning by two constraints: (i) data-generation mechanism is typically a known (or…
data-analysis-physics
Likelihood ratio: Λ = L(θ_1|x)/L(θ_0|x)
The likelihood ratio Λ = L_1/L_0 is the Neyman-Pearson 1933 most-powerful test statistic for simple-vs-simple hypothesis testing: given…
χ²(k) moments: E[χ²]=k, Var[χ²]=2k
The chi-square distribution χ²(k) with k degrees of freedom is the distribution of Σ_i^k Z_i² where Z_i ~ iid N(0,1). Probability density:…
Bootstrap distinct-fraction: 1 - (1-1/n)^n → 1 - 1/e
The bootstrap (Efron 1979) estimates sampling-distribution statistics by resampling n times with replacement from the original n data…
LR null: Λ(L_1=L_0) = 1
Sympy-exact witness of the likelihood-ratio null-state identity. Setup: Λ = L_1/L_0 as a two-symbol rational function. Identity: at L_1 =…
χ²(k=1) anchor: mean=1, variance=2
Sympy-exact witness of the k=1 chi-square distribution moments. Setup: E[χ²_k] = k, Var[χ²_k] = 2·k as one-symbol linear functions of k. …
Bootstrap distinct-limit: lim_{n→∞} [1-(1-1/n)^n] = 1-e^{-1}
Sympy-exact witness of the bootstrap distinct-fraction infinite-n limit. Setup: distinct-fraction = 1 - (1 - 1/n)^n as a function of n. …
Least-squares via SVD: beta-hat = V Sigma^+ U^T y; Frobenius-norm extremum
Least-squares via singular-value decomposition (SVD) framework (Eckart-Young 1936; Golub-Reinsch 1970). Setup: linear regression problem X…
MLE first-order optimality: d ln L/d theta = 0 at theta-hat; Fisher information
Maximum-likelihood-estimation framework (Fisher 1922 Phil Trans R Soc A 222, 309). Setup: given iid sample x_1, ..., x_n with density f(x;…
AIC for nested models: AIC = 2k - 2 ln L; Hahn-Banach extension on model functionals
Akaike Information Criterion framework (Akaike 1973 in Petrov-Csaki Information Theory). Setup: given a family of statistical models M_1,…
Theorem: trace(X X^T) - 13 = 0 at X = diag(2, 3) (SVD Frobenius-norm-squared)
Theorem (SVD-Frobenius-trace canonical): for X = diag(2, 3), the singular values are sigma_1 = 3, sigma_2 = 2 (sorted descending), and the…
Theorem: d/d mu [-(x - mu)^2/(2 sigma^2)] at mu = x equals 0 (MLE first-order)
Theorem (MLE-Gaussian-first-order canonical): for the Gaussian log-likelihood -(x - mu)^2/(2 sigma^2), differentiation gives d/d mu = (x -…
Theorem: Delta AIC - 2(k_2 - k_1) = 0 at L_1 = L_2 (equal-likelihood nested-model penalty)
Theorem (AIC-equal-likelihood canonical): when nested models M_1 subset M_2 have equal maximised likelihoods L_1 = L_2 (no improvement from…
Matched filter (Wiener 1949)
Wiener 1949 + North 1943 matched filter: SNR-optimal linear detector for known-template signal in white noise; basis of GW…
MCMC (Metropolis 1953 / Hastings 1970)
Metropolis-Rosenbluth 1953 + Hastings 1970: Markov-chain whose stationary distribution is target; ergodicity + detailed-balance; basis of…
HMC (Duane-Kennedy-Pendleton-Roweth 1987)
Duane et al 1987: Hamiltonian dynamics gives long-jump proposals avoiding random-walk diffusion; basis of NUTS sampler (Hoffman-Gelman…
Kalman filter (physics applications)
Kalman filter applied to GW source-localization, X-ray tracking, particle-trajectory reconstruction; recursive Bayes update for dynamical…
Bayesian model selection (Jeffreys 1939)
H Jeffreys 1939 Bayes-factor evidence: K = P(D|M1)/P(D|M2); MultiNest (Feroz 2008) + dynesty (Speagle 2020) nested-sampling for…
AIC / BIC (Akaike 1974, Schwarz 1978)
Akaike 1974 AIC = -2 log L + 2 k; Schwarz 1978 BIC = -2 log L + k log n; penalize parameter-count to avoid overfit; standard…
Least squares (Gauss 1809)
C Gauss 1809 + Legendre 1805 least-squares; modern modern foundational text + Markov-Gauss-theorem + 2024 ML-driven robust regression.
MLE (Fisher 1922)
R Fisher 1922 maximum-likelihood + 1925 Cramér-Rao; modern modern foundational text + Fisher-information + asymptotic-theory.
Bayesian (Laplace 1812)
P-S Laplace 1812 + Bayes 1763; modern modern foundational text + posterior-MCMC Metropolis-Hastings + Hamiltonian-MC + 2024…
Kalman filter (1960)
R Kalman 1960 + Bucy 1961 Kalman-Bucy linear-stochastic estimator; modern modern foundational + Apollo-guidance + 2024 deep-Kalman.
Monte-Carlo (Ulam 1947)
S Ulam-J von Neumann 1947 Monte-Carlo Manhattan-Project; modern modern foundational text + nested-sampling + sequential-MC + ML-MCMC.
ML for physics (Mehta 2019)
P Mehta 2019 ML-for-physics review; modern modern Hinton-RBM + RAVE + 2024 generative-physics 100k-cmpd protein generative AI.