inversegamma

The inverse gamma distribution is a continuous probability distribution that is the distribution of the reciprocal of a random variable that has a gamma distribution. It is often used in Bayesian statistics as a conjugate prior for the variance of a normal distribution. The probability density function of the inverse gamma distribution is defined for positive real numbers.

The inverse gamma distribution is characterized by two positive parameters, often denoted by alpha and beta. These parameters are sometimes referred to as the shape and scale parameters, respectively. The mean of an inverse gamma distribution exists if alpha > 1, and the variance exists if alpha > 2. As alpha increases, the distribution becomes more concentrated around its mean.

In Bayesian inference, the inverse gamma distribution is frequently employed as a prior distribution for the precision parameter of a normal distribution, which is the reciprocal of the variance. This choice of prior simplifies posterior calculations, as the posterior distribution for the precision also follows an inverse gamma distribution. The inverse gamma distribution is also related to other distributions, such as the log-gamma distribution.