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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 .
The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an "inverse" only in that, while the Gaussian describes a Brownian motion's level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a ...
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.
In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.
In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution [1]) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It is used in Bayesian inference as conjugate prior for the variance of the normal distribution. [2]
This distribution was first proposed by Étienne Halphen. [1] [2] [3] It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes. [4]
The class of normal-inverse Gaussian distributions is closed under convolution in the following sense: [9] if and are independent random variables that are NIG-distributed with the same values of the parameters and , but possibly different values of the location and scale parameters, , and ,, respectively, then + is NIG-distributed with parameters ,, + and +.
The method of inverse probability (assigning a probability distribution to an unobserved variable) is called Bayesian probability, the distribution of data given the unobserved variable is the likelihood function (which does not by itself give a probability distribution for the parameter), and the distribution of an unobserved variable, given ...