Mixture (probability)
Encyclopedia
In probability theory
and statistics
, a mixture is a combination of two or more probability distributions. The concept arises in two contexts:
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a mixture is a combination of two or more probability distributions. The concept arises in two contexts:
- A mixture defining a new probability distributionProbability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
from some existing ones, as in a mixture densityMixture densityIn probability and statistics, a mixture distribution is the probability distribution of a random variable whose values can be interpreted as being derived in a simple way from an underlying set of other random variables. In particular, the final outcome value is selected at random from among the...
. Here the main problem is to derive the theoretical properties of the new distribution. - A mixture used as a statistical modelStatistical modelA statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...
such as is often used for statistical classification.The model may represent the population from which observations arise as a mixture densityMixture densityIn probability and statistics, a mixture distribution is the probability distribution of a random variable whose values can be interpreted as being derived in a simple way from an underlying set of other random variables. In particular, the final outcome value is selected at random from among the...
, but the problem is that of a mixture modelMixture modelIn statistics, a mixture model is a probabilistic model for representing the presence of sub-populations within an overall population, without requiring that an observed data-set should identify the sub-population to which an individual observation belongs...
, in which a data classification hypothesis represents an overall distribution as a sum of separate distributions (representing separate populations) and the task is to infer from which population each observation arises.