Maximum Likelihood Estimation

In course

Description

This lecture covers the concept of Maximum Likelihood Estimation (MLE) in statistical inference, focusing on the estimation of parameters in regular parametric models. It explains the Cramér-Rao bound, the method of maximum likelihood, and the properties of MLE in both discrete and continuous cases. The lecture also discusses the Fisher information and the uniqueness of MLE in exponential families. The instructor provides examples of MLE for Bernoulli, exponential, and Gaussian distributions, highlighting the importance of equivariance in MLE. The lecture concludes with a general proposition on the uniqueness of MLE in one-parameter exponential families.

Instructors (2)

Official source

https://mediaspace.epfl.ch/media/0_n28o694i

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Related lectures (79)

Related concepts (120)

Maximum likelihood estimation

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

Fisher information

In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher (following some initial results by Francis Ysidro Edgeworth).

Statistics

Statistics (from German: Statistik, () "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal".

Statistical inference

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population.

January 9

681 – Twelfth Council of Toledo: King Erwig of the Visigoths initiates a council in which he implements diverse measures against the Jews in Spain. 1127 – Jin–Song Wars: Invading Jurchen soldiers from the Jin dynasty besiege and sack Bianjing (Kaifeng), the capital of the Song dynasty of China, and abduct Emperor Qinzong of Song and others, ending the Northern Song dynasty. 1349 – The Jewish population of Basel, believed by the residents to be the cause of the ongoing Black Death, is rounded up and incinerated.