Lecture

Statistical Models: Families and Transformations

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In course

MATH-240: Statistics

Ce cours donne une introduction au traitement mathématique de la théorie de l'inférence statistique en utilisant la notion de vraisemblance comme un thème central. ...

Description

This lecture covers statistical models, focusing on families of distributions and transformations. It discusses exponential families, natural vs. usual forms, examples like the binomial and Gaussian families, and probability models. The lecture also explores probability transformations, including discrete and continuous cases, affine transformations, and multidimensional transformations. Applications such as convolutions of densities and sums of normal random variables are presented, along with relevant corollaries.

Instructors (2)

Victor Panaretos

Curriculum vitae Background & EducationPublications & PreprintsHonours & AwardsEditorial ActivitiesInvited Talks & Seminars Teaching & PhD PhD Students Ulysse Naepels, Nikitas Georgakis, Ho Yun Past EPFL PhD Students Andrea Kraus, Shahin Tavakoli, Mikael Kuusela, Yoav Zemel, Marie-Hélène Descary, Valentina Masarotto, Tomas Rubin, Tomas Masák, Laya Ghodrati, Kartik Waghmare, Leonardo Santoro Past EPFL PhD Students as codirector Matthieu Simeoni, Ophelia Miralles, Sonia Alouini, Timmy Rong Tian Tse, Krali M

Yoav Zemel

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Related concepts (32)

Normal distribution

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem.

Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.

Log-normal distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values.

Bayesian probability

Bayesian probability (ˈbeɪziən or ˈbeɪʒən ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown.

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