Compound probability distribution

compound distributionmixturecompoundingcompoundcompound distributionscompounded distributionmixture model
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.wikipedia
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Mixture distribution

mixture densitymixturedensity mixture
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.
More general cases (i.e. an uncountable set of component distributions), as well as the countable case, are treated under the title of compound distributions.

Overdispersion

underdispersionover-dispersionoverdispersed
Compound distributions are useful for modeling outcomes exhibiting overdispersion, i.e., a greater amount of variability than would be expected under a certain model.
In the case of count data, a Poisson mixture model like the negative binomial distribution can be proposed instead, in which the mean of the Poisson distribution can itself be thought of as a random variable drawn – in this case – from the gamma distribution thereby introducing an additional free parameter (note the resulting negative binomial distribution is completely characterized by two parameters).

Posterior predictive distribution

prior predictive distributionpredictive uncertainty quantification
This gives a posterior predictive distribution.
The prior predictive distribution is in the form of a compound distribution, and in fact is often used to define a compound distribution, because of the lack of any complicating factors such as the dependence on the data \mathbf{X} and the issue of conjugacy.

Marginal distribution

marginal probabilitymarginalmarginals
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"). 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 and \theta is given by, and the compound results as its marginal distribution:

Bayesian inference

BayesianBayesian analysisBayesian method
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.
Both types of predictive distributions have the form of a compound probability distribution (as does the marginal likelihood).

Gamma distribution

gammagamma distributedGamma variate
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.
The compound distribution, which results from integrating out the inverse-scale, has a closed form solution, known as the compound gamma distribution.

Beta-binomial distribution

Beta-binomial modelbeta-binomialBeta-binomial Emission Densities
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.
This fact leads to an analytically tractable compound distribution where one can think of the p parameter in the binomial distribution as being randomly drawn from a beta distribution.

Negative binomial distribution

negative binomialGamma-Poisson distributioninverse binomial distribution
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.
The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution.

Exponential family

exponential familiesnatural parameternatural parameters
Compound distributions derived from exponential family distributions often have a closed form.
In general, distributions that result from a finite or infinite mixture of other distributions, e.g. mixture model densities and compound probability distributions, are not exponential families.

Dirichlet-multinomial distribution

Multivariate Pólya distributionDirichlet-multinomialDirichlet-multinomial model
It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector, and an observation drawn from a multinomial distribution with probability vector p and number of trials n.

Gibbs sampling

Gibbs samplercollapsed Gibbs samplingGibbs
distributions p(\theta) as well as p(x|\theta) and then utilize these to perform collapsed Gibbs sampling to generate samples from p(x).
The distribution over a variable A that arises when collapsing a parent variable B is called a compound distribution; sampling from this distribution is generally tractable when B is the conjugate prior for A, particularly when A and B are members of the exponential family.

Student's t-distribution

Student's ''t''-distributiont-distributiont''-distribution
The distribution is thus the compounding of the conditional distribution of \mu given the data and \sigma^2 with the marginal distribution of \sigma^2 given the data.

Expectation–maximization algorithm

expectation-maximization algorithmEM algorithmexpectation maximization
Parameter estimation (maximum-likelihood or maximum-a-posteriori estimation) within a compound distribution model may sometimes be simplified by utilizing the EM-algorithm.

Beta prime distribution

Beta primecompound gamma distributionGeneralized Beta Prime
It is so named because it is formed by compounding two gamma distributions:

Lomax distribution

Lomax
The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution.

Exponentially modified Gaussian distribution

ExGaussian distributionEMG distributionExponentially modified Gaussian
The distribution is a compound probability distribution in which the mean of a normal distribution varies randomly as a shifted exponential distribution.

Probability and statistics

probability, statistics
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.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
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.

Random variable

random variablesrandom variationrandom
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.

Probability density function

probability densitydensity functiondensity
Its probability density function is given by:

Joint probability distribution

joint distributionjoint probabilitymultivariate distribution
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 and \theta is given by, and the compound results as its marginal distribution:

Variance

sample variancepopulation variancevariability
A compound distribution H resembles in many ways the original distribution F that generated it, but typically has greater variance, and often heavy tails as well.

Heavy-tailed distribution

heavy tailsheavy-tailedheavy tail
A compound distribution H resembles in many ways the original distribution F that generated it, but typically has greater variance, and often heavy tails as well.

Support (mathematics)

supportcompact supportcompactly supported
The support of H is the same as the support of the F, and often the shape is broadly similar as well.