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TL;DR:

  • I have added a tag wiki and an excerpt to . The excerpt notes that these are also known as "mixtures", but that that term also has other uses.
  • I have included a warning in the tag wiki excerpt for . It now reads:

    A mixture distribution is one that is written as a convex combination of other distributions. Use the "compound-distributions" tag for "concatenations" of distributions (where a parameter of a distribution is itself a random variable).

    I will link to this meta question in the tag wiki.

  • I will go through threads with the tag in the next days and retag them as as warranted.


Explanation:

The term "mixture" has two distinct meanings in our context:

  • A mixture distribution is a convex combination of two or more underlying distributions with associated probabilities $p_i$ that sum to one. A realization of the mixture distribution comes with probability $p_i$ from the $i$-th underlying distribution. Examples include:
    • The mixture of two or more normal distributions with different means and/or variances: s.
    • The mixture of a point mass at zero with another (discrete) distribution: .
  • A mixture may also be a "concatenation" of two distributions (), where our realizations are distributed according to some parameterized distribution $X\sim F_\theta$ where the parameter $\theta$ itself is a random variable, $\theta\sim G$. So, to draw an $x$, you first draw a $\theta\sim G$ and then, with this $\theta$, draw $x\sim F_\theta$. $G$ is called the mixing distribution. Examples include:
    • The distribution arises naturally as a Poisson distribution whose parameter is itself gamma distributed: it is a "Poisson-gamma mixture".

It does not help that even when people discuss compound distributions as such, they almost invariably call the resulting compound distribution an "$F$-$G$ mixture", as in the negbin case (not, e.g., an "$F$-$G$ compound"). Ah well.


Our tag refers exclusively to the first meaning above. Here is its tag wiki excerpt:

A mixture distribution is one that is written as a convex combination of other distributions.

The overloaded nomenclature results in people tagging questions as when they are actually asking about mixtures in the second sense above, i.e., about .

The "disambiguation" section of the tag wiki only disambiguates it from mixed models.


Any thoughts, comments or improvements on the tag wikis would be more than welcome.

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    $\begingroup$ Agreed. Less confusion is better. $\endgroup$ – Carl May 2 at 9:49
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    $\begingroup$ A mixture of two normal distributions might be regarded as a compound of normal & Bernoulli distributions, in general or not depending on what you allow to count as parametrization. If you want to make a distinction, I suspect it's that compounds are mixtures of "an infinite" number of" distributions, or suchlike. $\endgroup$ – Scortchi - Reinstate Monica May 2 at 10:21
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The two meanings don't seem all that distinct. If, in the definition of the 1st sense of "mixture", you take probabilities as probability densities & sums as integrals, you get a mixture in the 2nd sense; if, in the definition of the 2nd sense, you allow $\theta$ to index a countable set of distributions, you get a mixture in the first sense. (And there seems to be a gap—what do you call it when $\theta$ indexes an uncountable set of distributions that don't constitute a parametric family?) At any rate the crux of the distinction needs to be made more explicit.

Even if compound distributions are more properly viewed as an extension of mixture distributions rather than as a sub-class, I wonder whether it mightn't be better to let mixture cover the general idea, & compound distribution the particular case—practically, to refrain from removing the mixture tag when adding the compound distribution tag.

† Wikipedia doesn't help: the article on mixture distributions defines compound distributions as those where $\theta$ indexes a set of uncountable distributions; while the article on compound distributions defines them as you do.

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  • $\begingroup$ I appreciate your answer, but I'm downvoting, because I do not believe this is a good approach. Yes, I agree that the two senses of "mixture" are not all that distinct, and on a sufficiently abstract level, the two notions pretty much coincide. (I'm a mathematician by training, I'm all for unifying concepts by abstraction.) However, most people have very different toolsets for and very different questions about the two problems on a less abstract level. ... $\endgroup$ – Stephan Kolassa May 3 at 12:47
  • $\begingroup$ ... Judging from the questions with the mixture tag, conflating them with compound-distributions will not be helpful - neither for the original askers, nor for anyone who later comes along and searches by tags for particular questions that may be helpful, either by answering their problem or as duplicates. $\endgroup$ – Stephan Kolassa May 3 at 12:48

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