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Sorry if my question is clear to most of you. As a mathematics background, I really just start working with the mixture and would like to understand it in a clear way. Mixture dependencies are when the dependencies structures between two variables come from different distribution. For example, if I have an air pollution data and would like to model the relationship between air pollution within cities in one state of a country and the cities over the country. Then, this data will be a mixture since we have two different type of dependencies. If not, could someone help me with this using an example?

many thanks.

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Mixture dependencies are when the dependencies structures between two variables come from different distribution.

No. The finite mixture distribution is a distribution of a single random variable, that is defined in terms of a "mixture" of other distributions. The mixture probability density function $g$ is called a mixture of distributions $f_1,\dots,f_m$ when

$$ g(x) = \sum_{k=1}^m \pi_k \, f_k(x; \theta_k) $$

where $\theta_1,\dots,\theta_m$ are the parameters and $\pi_1,\dots,\pi_m$ such that $0 \le \pi_k \le 1$ and $\sum_k \pi_k = 1$ are mixing proportions.

To give a real-life example, say that you are looking at the distribution of heights of teenagers. The distribution is a mixture of distributions of heights of boys and girls (that differ). If you knew who are the boys and girls in your data, you could model the distributions of their heights separately, or preferably in a hierarchical model, but if you are given the unlabeled data, then possibly two-component mixture would fit your data better, then "single", unimodal distribution.

Your air pollution example sounds rather like an example of mixed effects model.

Tim
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