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Concept
Compound probability distribution
Summary
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 is distributed according to some parametrized distribution with an unknown parameter that is again distributed according to some other distribution . The resulting distribution is said to be the distribution that results from compounding with . The parameter's distribution is also called the mixing distribution or latent distribution. Technically, the unconditional distribution results from marginalizing over , i.e., from integrating out the unknown parameter(s) . Its probability density function is given by:
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 and is given by
and the compound results as its marginal distribution:
If the domain of is discrete, then the distribution is again a special case of a mixture distribution.
The compound distribution will depend on the specific expression of each distribution, as well as which parameter of is distributed according to the distribution , and the parameters of will include any parameters of that are not marginalized, or integrated, out.
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