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In general, the marginal probability distribution of X can be determined from the joint probability distribution of X and other random variables. If the joint probability density function of random variable X and Y is , (,), the marginal probability density function of X and Y, which defines the marginal distribution, is given by: =, (,)
The distributions of each of the component random variables are called marginal distributions. The conditional probability distribution of X i {\displaystyle X_{i}} given X j {\displaystyle X_{j}} is the probability distribution of X i {\displaystyle X_{i}} when X j {\displaystyle X_{j}} is known to be a particular value.
Representing a joint distribution as univariate margins plus copulas allows the separation of the problems of estimating univariate distributions from the problems of estimating dependence. This is handy in as much as univariate distributions in many cases can be adequately estimated from data, whereas dependence information is roughly unknown ...
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Copulas are used to describe/model the dependence (inter-correlation) between random variables. [1]
One can compute this directly, without using a probability distribution (distribution-free classifier); one can estimate the probability of a label given an observation, (| =) (discriminative model), and base classification on that; or one can estimate the joint distribution (,) (generative model), from that compute the conditional probability ...
Joint and marginal distributions of a pair of discrete random variables, X and Y, dependent, thus having nonzero mutual information I(X; Y). The values of the joint distribution are in the 3×4 rectangle; the values of the marginal distributions are along the right and bottom margins.
This rule allows one to express a joint probability in terms of only conditional probabilities. [4] The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
Mutual information is a measure of the inherent dependence expressed in the joint distribution of and relative to the marginal distribution of and under the assumption of independence. Mutual information therefore measures dependence in the following sense: I ( X ; Y ) = 0 {\displaystyle \operatorname {I} (X;Y)=0} if and only if X ...