Editing Mixture model (section)

=== Definition ===
Consider a mixture of parametric distributions of the same class. Let

<math display="block">J = \{f(\cdot ; \theta):\theta\in\Omega\}</math>

be the class of all component distributions. Then the [[convex hull]] {{math|''K''}} of {{math|''J''}} defines the class of all finite mixture of distributions in {{math|''J''}}:

<math display="block">K = \left\{p(\cdot):p(\cdot)=\sum_{i=1}^n a_i f_i(\cdot ; \theta_i), a_i>0, \sum_{i=1}^n a_i=1, f_i(\cdot ; \theta_i)\in J\ \forall i,n\right\}</math>

{{math|''K''}} is said to be identifiable if all its members are unique, that is, given two members {{math|''p''}} and {{math|''p&prime;''}} in ''K'', being mixtures of {{mvar|k}} distributions and {{mvar|k&prime;}} distributions respectively in {{math|''J''}}, we have {{math|1=''p'' = ''p&prime;''}} if and only if, first of all, {{math|1=''k'' = ''k&prime;''}} and secondly we can reorder the summations such that {{math|1=''a''<sub>''i''</sub> = ''a''<sub>''i''</sub>&prime;}} and {{math|1=''f''<sub>''i''</sub> = ''f''<sub>''i''</sub>&prime;}} for all {{mvar|i}}.