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What is the difference between mixture and hierarchical models?

Are they of the same nature with different names or they are totally different things?

If there are any references, I will be happy to know.

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Terminology is not as standardized as one may whish, so what I understand under these terms may not be what others understand under these terms.

I understand under a mixture model a model that posits that posits a mixture of "brand name distributions" for the dependent variable. For example a discrete mixture of normals assumes that a person is drawn with a to be estimated probability from one normal distribution and with another probability form another normal distribution, etc., for a given number of groups. Notice that in this model we do not know which person belongs to which group

Under a hierarchical model I understand that we have observations that are nested in groups. For example: students are nested in classrooms, which are nested in schools, which are nested in countries. Here membership of the groups is known and part of the data.

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  • $\begingroup$ Yes. I too understand that mixture means a (convex) linear combination of known models while hierachicials means the parameters' distributions. I were just confused when I read some paper using both interchangebly. Is there such a freedom of use of these two terms in statisticans' world? $\endgroup$
    – Henry.L
    Jan 25, 2016 at 20:09
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    $\begingroup$ It depends upon your frame of reference. To extend the school example, if you were to select a random student without knowledge of the nesting, or predict the performance of a student at a school without knowledge of the classroom, the distribution of the variable of interest would be a mixture, but the appropriate model would still be hierarchical. However, given knowledge of the entire hierarchy, the performance might or might not be a mixture, depending upon how individual performance at the classroom was modeled (you could imagine it was a sex-based mixture but we don't know sex.) $\endgroup$
    – jbowman
    Jan 25, 2016 at 21:07
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    $\begingroup$ My guess would be that @jbowman is correct, but yes the terminology is far from standardized in statistics. So you should not rely on them, and instead, when writing articles, explicitly define how you use those terms. Statistics plays more of a supporting role, which has led to groups that are more alligned with one substantive discipline (econometrics, psychometrics, ...). This is good in the sense that that way tools are developed that are actually used, but bad in that it has led to a terminological chaos. I think it is a price worth paying, but it is a price none the less. $\endgroup$ Jan 26, 2016 at 9:00

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