Density mixes

In Statistical, one calls density mixes , or law mixes a function of density which is resulting from a linear Combinaison several functions of density.

By taking a function $f (X, \ theta)$ , density of a Random variable $x$ parameterized by $\ theta$ . For example, if F is a normal Loi, then $\ theta$ is consisted of the Moyenne and the Variance. If one calls $\ theta_1, \ dowries, \ theta_g$ a family of parameters and $\ pi_1, \ dowries, \ pi_g$ a family of scalars such as

$\ sum_ {k=1} ^g \ pi_k=1$ ,

then, the function $g$ defined by

$g (X, \ theta_1, \ dowries, \ theta_g) = \ sum_ {k=1} ^g \ pi_kf (X, \ theta_k)$

the function of density of a law is a function mixes with $g$ component.

One can also extend this definition if the number of the components is infinite. By considering a unit $\ Omega$ of parameters, if one has

$\ int_ \ Omega \ pi \ left (\ theta \ right) D \ theta=1,$

then, the function

$g \ left (X, \ Omega \ right) = \ int_ \ Omega \ pi (\ theta) F (X, \ theta) D \ theta$

is a density mixes.

For example, the following image represents the function of density of a Gaussian mixture to two components of dimension 2

Density mixes

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