Compound probability distribution

In probability theory, a compound distribution is the probability distribution that results from assuming that a random variable is distributed according to some parameterized distribution $F$ with an unknown parameter $\theta$ that is distributed according to some other distribution $G$, and then determining the distribution that results from marginalizing over $G$ (i.e. integrating the unknown parameter out). The resulting distribution $H$ is said to be the distribution that results from compounding $F$ with $G$. Typically, $G$ is assumed to be the conjugate prior of $F$.

Examples:

• Compounding a Gaussian distribution with mean distributed according to another Gaussian distribution yields a Gaussian distribution.
• Compounding a Gaussian distribution with precision (inverse variance) distributed according to a gamma distribution yields a three-parameter Student's t distribution.
• Compounding a binomial distribution with probability of success distributed according to a beta distribution yields a beta-binomial distribution.
• Compounding a multinomial distribution with probability vector distributed according to a Dirichlet distribution yields a multivariate Polya distribution, also known as a Dirichlet compound multinomial distribution.

Note that the support of the resulting compound distribution $H$ is the same as the support of the original distribution $F$. For example, a beta-binomial distribution is discrete just as the binomial distribution is (however, its shape is similar to that of a beta distribution). The variance of the compound distribution $H$ is typically greater than the variance of the original distribution $F$. The parameters of $H$ include the parameters of $G$ and any parameters of $F$ that are not marginalized out. For example, the beta-binomial distribution includes three parameters, a parameter $n$ (number of samples) from the binomial distribution and shape parameters $\alpha$ and $\beta$ from the beta distribution.

A related but slightly different concept of "compound" occurs in the compound Poisson distribution, which is actually a set of distributions defined by a parameter $J$ that is itself a distribution. These distributions result from considering a set of independent identically-distributed random variables distributed according to $J$ and asking what the distribution of their sum is, if the number of variables is itself an unknown random variable $N$ distributed according to a Poisson distribution and independent of the variables being summed. In this case the random variable $N$ is marginalized out much like $\theta$ above is marginalized out.