- I have added a tag wiki and an excerpt to compound-distributions. 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 mixture. 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.
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:
- A mixture may also be a "concatenation" of two distributions (compound-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 negative-binomial 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 mixture 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 "disambiguation" section of the mixture tag wiki only disambiguates it from mixed models.
Any thoughts, comments or improvements on the tag wikis would be more than welcome.