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In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.
The closely related inverse-gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution. If α is a positive integer, then the distribution represents an Erlang distribution; i.e., the sum of α independent exponentially distributed random variables, each of which has a mean of θ.
In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters .
Usually, the inverse gamma function refers to the principal branch with domain on the real interval [, +) and image on the real interval [, +), where = … [2] is the minimum value of the gamma function on the positive real axis and = = … [3] is the location of that minimum.
The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. = , = . The reason for the usefulness of this characterization is that in Bayesian statistics the inverse gamma distribution is the conjugate prior distribution of the variance of a Gaussian ...
In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance .
Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible and since the GH distribution can be obtained as a normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution, Barndorff-Nielsen and Halgreen showed the GH distribution is infinitely divisible as well. [3]
In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices. [1] It is a more general version of the inverse Wishart distribution, and is used similarly, e.g. as the conjugate prior of the covariance matrix of a multivariate normal distribution or matrix normal distribution.