Compound probability distribution
In probability and statistics, a compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with 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 is the result of marginalizing over the latent random variable representing the parameter of the parametrized distribution.
Definition
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. 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.
Properties
A compound distribution resembles in many ways the original distribution that generated it, but typically has greater variance, and often heavy tails as well. The support of is the same as the support of the, and often the shape is broadly similar as well. The parameters of include any parameters of or that have not been marginalized out.The compound distribution's first two moments are given by
and
.
Applications
Testing
Distributions of common test statistics result as compound distributions under their null hypothesis, for example in Student's t-test, or in the F-test.Overdispersion modeling
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.Bayesian inference
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 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.Computation
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 as well as and then utilize these to perform collapsed Gibbs sampling to generate samples from.
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 within a compound distribution model may sometimes be simplified by utilizing the EM-algorithm.
Examples
- Gaussian scale mixtures:
- * Compounding a normal distribution with variance distributed according to an inverse gamma distribution yields a non-standardized Student's t-distribution. This distribution has the same symmetrical shape as a normal distribution with the same central point, but has greater variance and heavy tails.
- * Compounding a Gaussian distribution with variance distributed according to an exponential distribution yields a Laplace distribution.
- * Compounding a Gaussian distribution with variance distributed according to an exponential distribution whose rate parameter is itself distributed according to a gamma distribution yields a Normal-exponential-gamma distribution.
- * Compounding a Gaussian distribution with standard deviation distributed according to a inverse uniform distribution yields a Slash distribution.
- other Gaussian mixtures:
- * Compounding a Gaussian distribution with mean distributed according to another Gaussian distribution yields a Gaussian distribution.
- * Compounding a Gaussian distribution with mean distributed according to a shifted exponential distribution yields an exponentially modified Gaussian distribution.
- Compounding a binomial distribution with probability of success distributed according to a beta distribution yields a beta-binomial distribution. It possesses three parameters, a parameter from the binomial distribution and shape parameters and from the beta distribution.
- Compounding a multinomial distribution with probability vector distributed according to a Dirichlet distribution yields a Dirichlet-multinomial distribution.
- Compounding a Poisson distribution with rate parameter distributed according to a gamma distribution yields a negative binomial distribution.
- Compounding an exponential distribution with its rate parameter distributed according to a gamma distribution yields a Lomax distribution.
- Compounding a gamma distribution with inverse scale parameter distributed according to another gamma distribution yields a three-parameter beta prime distribution.
- Compounding a half-normal distribution with its scale parameter distributed according to a Rayleigh distribution yields an exponential distribution. This follows immediately from the Laplace distribution resulting as a normal scale mixture; see above. The roles of conditional and mixing distributions may also be exchanged here; consequently, compounding a Rayleigh distribution with its scale parameter distributed according to a half-normal distribution also yields an exponential distribution.
- A Gamma - distributed random variable whose scale parameter θ again is uniformly distributed marginally yields an exponential distribution.