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DEFINITION of 'Compound Probability'
Compound Probability can be defined as a mathematical term relating to the likeliness of two independent events occurring and the compound probability is equal to the probability of the first event multiplied by the probability of the second event.
The compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parameterized distribution with an unknown parameter θ or parameter vector θ. This is distributed according to some other distribution G with hyper parameter α, and later on determining the distribution that results from marginal distribution over G (i.e. integrating the unknown parameter(s) out) and the resulting distribution H is said to be the distribution that results from compounding F with G and the Bayesian inference, the distribution G is described as a conjugate prior of F and it is expressed mathematically for a scalar data point with scalar parameter and hyperparameter as follows:
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The same formula is applicable with some or all of the variables are vectors and in this case for a vector data point with vector parameters and hyper parameters are represented as follows
A compound distribution H resembles in many ways the original distribution F that generated it, but typically has greater variance and the support of H is the same as the support of the F. The shape is broadly similar as well and the parameters of H include the parameters of G and any parameters of F that are not marginalized out.
Compounding a normal distribution with variance distributed according to an inverse gamma distribution with precision distributed as a gamma distribution yields a non-standardized student t-distribution. This distribution has a similar symmetrical shape as a normal distribution with the same central point but has greater variance as well as the heavy tail.
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Compounding a binomial distribution with probability of success distributed according to beta distribution yields a beta-binomial distribution as this distribution is a discrete distribution just as the binomial distribution with support for integers between 0 and n. There are three parameters and the parameter n which is the number of samples from the binomial distribution and shape parameter and from, Alpha α and Beta β distribution and the shape are the same as a binomial distribution when α Alpha, β Beta is high. This indicates very high certainty that the prior probability is quite near a specific location.
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