# 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.

• Agreed. Less confusion is better.
– Carl
May 2, 2020 at 9:49
• 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. May 2, 2020 at 10:21

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.
† 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.