In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables.If the parameter is a scale parameter, the resulting mixture is also called a scale mixture.
The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").
A compound probability distribution is the probability distribution that results from assuming that a random variable
X
F
\theta
G
H
F
G
G
H
G
\theta
pH(x)={\displaystyle\int\limitspF(x|\theta)pG(\theta)\operatorname{d}\theta}
The same formula applies analogously if some or all of the variables are vectors.
From the above formula, one can see that a compound distribution essentially is a special case of a marginal distribution: The joint distribution of
x
\theta
p(x,\theta)=p(x|\theta)p(\theta)
{stylep(x)=\intp(x,\theta)\operatorname{d}\theta}
\theta
The compound distribution
H
F
G
H
G
H
F
The compound distribution's first two moments are given by the law of total expectation and the law of total variance:
If the mean of
F
G
\mu
\sigma2
\operatorname{E}H[X]=\operatorname{E}G[\theta]=\mu
\operatorname{Var}H(X)=\operatorname{Var}F(X|\theta)+\operatorname{Var}G(Y)=\tau2+\sigma2
\tau2
F
let
F
G
f(x|\theta)=pF(x|\theta)
g(\theta)=pG(\theta)
h(x)
H
l{F}
l{G}
\operatorname{E}H[X]=\mu
The variance of
H
2] | |
\operatorname{E} | |
H[X |
-
2 | |
(\operatorname{E} | |
H[X]) |
\intFx2f(x\mid\theta)
2\mid | |
dx=\operatorname{E} | |
F[X |
\theta]=\operatorname{Var}F(X\mid\theta)+(\operatorname{E}F[X\mid\theta])2
\intG\theta2
2 | |
g(\theta)d\theta=\operatorname{E} | |
G[\theta |
]=\operatorname{Var}G(\theta)+
2 | |
(\operatorname{E} | |
G[\theta]) |
Distributions of common test statistics result as compound distributions under their null hypothesis, for example in Student's t-test (where the test statistic results as the ratio of a normal and a chi-squared random variable), or in the F-test (where the test statistic is the ratio of two chi-squared random variables).
Compound distributions are useful for modeling outcomes exhibiting overdispersion, i.e., a greater amount of variability than would be expected under a certain model. For example, count data are commonly modeled using the Poisson distribution, whose variance is equal to its mean. The distribution may be generalized by allowing for variability in its rate parameter, implemented via a gamma distribution, which results in a marginal negative binomial distribution. This distribution is similar in its shape to the Poisson distribution, but it allows for larger variances. Similarly, a binomial distribution may be generalized to allow for additional variability by compounding it with a beta distribution for its success probability parameter, which results in a beta-binomial distribution.
Besides ubiquitous marginal distributions that may be seen as special cases of compound distributions, in Bayesian inference, compound distributions arise when, in the notation above, F represents the distribution of future observations and G is the posterior distribution of the parameters of F, given the information in a set of observed data. This gives a posterior predictive distribution. Correspondingly, for the prior predictive distribution, F is the distribution of a new data point while G is the prior distribution of the parameters.
Convolution of probability distributions (to derive the probability distribution of sums of random variables) may also be seen as a special case of compounding; here the sum's distribution essentially results from considering one summand as a random location parameter for the other summand.[1]
Compound distributions derived from exponential family distributions often have a closed form.If analytical integration is not possible, numerical methods may be necessary.
Compound distributions may relatively easily be investigated using Monte Carlo methods, i.e., by generating random samples. It is often easy to generate random numbers from the distributions
p(\theta)
p(x|\theta)
p(x)
A compound distribution may usually also be approximated to a sufficient degree by a mixture distribution using a finite number of mixture components, allowing to derive approximate density, distribution function etc.
Parameter estimation (maximum-likelihood or maximum-a-posteriori estimation) within a compound distribution model may sometimes be simplified by utilizing the EM-algorithm.[2]
p
X
E[X]
X
n
\alpha
\beta
The notion of "compound distribution" as used e.g. in the definition of a Compound Poisson distribution or Compound Poisson process is different from the definition found in this article. The meaning in this article corresponds to what is used in e.g. Bayesian hierarchical modeling.
The special case for compound probability distributions where the parametrized distribution
F