Notes

Known persons in statistics learning

• Judea Pearl, Bayesian networks, and causality
• Geoffrey Hinton, backpropagation, deep belief networks
• Michael Jordan, popularize probability graphical models
• Yann Lecun, convolutional neural networks
• Yoshua Bengio, neural machine translation

Group:

• magma, nonempty set with a binary operation closed
• semigroup, magma with the binary operation associative
• inverse semigroup, semigroup with inverse
• monoid, semigroup with one identity element
• group, monoid with inverse (equivalently, inverse semigroup with one indentity element)
• Albelian group, group with the binary operation commutative
• quasigroup, magma where division is always possible
• loop, quasigroup with one identity element

Measure:

• sigma algebra: Sigma is a collection of subsets of X, such that, contains X as an element, closed under complement, closed under countable unions
• measurable space (Borel space): (X, Sigma), with sets in Sigma called measurable sets
• measure: mu: Sigma -> R U {-infty, +infty}, with nonnegativity, null empty set, countable additivity
• measure space: (X, Sigma, mu), or measurable space (X, Sigma) equipped with a measure mu
• probability measure, probability space (Omega, Sigma, P): measure P(X) = 1
• measurable map: f: (X, Sigmax) -> (Y, Sigmay), if the pre-image f belongs to Sigmax, for every element in Sigmay
• measurable function: f: (X, Sigmax) -> (R U {-infty, +infty}, Borel sets)
• Random variable is a measurable map (limited measurable function?)

Limited and Bounded:

• almost surely limited function: mu(|f| = +infty) = 0
• almost syrely bounded function: there is a positive constant M, such that mu(|f| > M) = 0

Gamma function/distribution:

• function: Gamma(z) = int(0, +\infty) x^{z-1} e^{-x} dx
• prabability density function: f(x) = beta^alpha / Gamma(alpha) x^{alpha - 1} e^{-beta x}, with mean alpha/beta, variance alpha/beta^2

Statistics:

• (statistical) population: a set of similar items or events of interest, X ~ P
• (statistical) sample: a set of individuals selected from a statistical population by a defined procedure, X_1, X_2, …, X_n ~ P
• sample points or observations: elements of a sample, x_1, x_2, …, x_n; sample size, n
• i.i.d.: a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent
• For a statistical parameter, theta, one sample outputs one point estimate; multiple samples ouput multiple point estimates, then interval estimates
• confidence interval with a confidence level C (C = 1 - alpha, with alpha significance level)
• statistical hypothesis inference testing is SHIT! misuse of p-value
• odds: P(X=1) / P(X=0), log-odds, likelihood ratio

Latin Abbreviation:

• etc. = ect cetera = and the rest
• i.e. = id est = that is = also known as = a.k.a.
• e.g. = exempli gratia = for example