nLab statistics

This entry (apart from a few remarks) will be mainly about general statistics and mathematical statistics. For more about physical applications in statistical mechanics and probabilistic interpretation of quantum mechanics, see there. There is also a technical notion of statistic (singular).

Idea

Statistics studies the analysis of collections of random or sample data, and the probabilistic likelihood of various inferences on the basis of these data, as well as the mathematical regularities in large ensembles of occurrences of such data.

In physics, statistics also pertains to the behaviour of large ensembles of particles. For identical particles, this is the subject of particle statistics and for general systems the subject of statistical mechanics.

Mathematical statistics is based on probability theory. Most of the standard formalism uses measure theory as used in probability. Statistical mechanics in addition heavily uses ergodic theory.

Definitions

A statistical model is a measurable function $P : \Theta \to \Delta(X)$, where $X$ and $\Theta$ are measurable spaces and $\Delta(X)$ is the simplex of probability measures over $X$.

Related entries

• probability, probability theory

• statistical inference

• statistical significance

• stochastic process,

• statistical mechanics

• time series,

• Giry monad,

• Bayesian reasoning

• conditional expectation

• information geometry

• Bayesian interpretation of quantum mechanics,

Literature

• Michael Schmitt, Statistics for theorists, quick 3-lecture intro for theoretical physicists at TASI 2020 (each around 1 and half hours) lec1 mp4 slides pdf, lec2 mp4 pdf, lec3 mp4 pdf

• wikipedia: statistics, statistic (singular!), list of statistical packages, descriptive statistics, inferential statistics, data mining, statistical theory, estimation theory, statistical significance, regression analysis, time series analysis, misuse of statistics, applied statistics, Bayesian inference, multivariate statistics, maximal likelihood estimation, Fisher information

• Peter McCullagh, What is a statistical model?, Ann. Statist. 30:5 (2002), 1225-1310 euclid MR1936320 doi – on applying category theory to describe statistical models.

• Lior Pachter, Bernd Sturmfels, Tropical geometry of statistical models, PNAS 101 no. 46, 16132–16137, doi

category: probability, physics