Probabilistic mixture models

In many practical applications, the familiar statistical distributions are not enough when distribution of the statistical data is two, three modal and make the mixture distributions become important. Nowadays, entropy optimization distributions (EOD) are known to be well besides mixture distributions. From this point of view, in this thesis by the aim of finding more appropriate models EOD probabilistic mixture models and the probabilistic mixture of EOD and familiar statistical distributions are the investigation topic. In present thesis, firstly, alternative proof of the known identifiability theorem for the identifiability of the finite mixture distribution is given by using Lebesgue-Stieltjes integral. Then, finite mixture of non-parametric EOD are proved to be identifiable; in order to present the probabilistic mixture of EOD and familiar statistical distributions, the non-parametric EOD are defined to be parametrized of one/two parameter. By choosing special moment functions set, the probabilistic mixture models of parametrized EOD are proved to be identifiable. New LGE, GWE, LWE, LGWE probabilistic mixture models consist of the double, triple union of the EOD distributions and Lognormal, Weibull and Gamma distributions family are proposed and proved to be identifiable. Furthermore, finite mixture distributions of the union of the logarithimic series distributions family and discrete rectangular distributions family is proved to be identifiable. The proposed non-parametric EOD probabilistic mixture models are applied on image processing and error distribution problems. Keywords: Probabilistic mixture model, Identifiability, Entropy optimization distributions, Parametrization, Union of different parametric family.