Normal-inverse-gamma distribution

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. Suppose has a normal distribution with mean and variance , where has an inverse-gamma distribution. Then has a normal-inverse-gamma distribution, denoted as ( is also used instead of ) The normal-inverse-Wishart distribution is a generalization of the normal-inverse-gamma distribution that is defined over multivariate random variables. For the multivariate form where is a random vector, where is the determinant of the matrix . Note how this last equation reduces to the first form if so that are scalars. It is also possible to let in which case the pdf becomes In the multivariate form, the corresponding change would be to regard the covariance matrix instead of its inverse as a parameter. Given as above, by itself follows an inverse gamma distribution: while follows a t distribution with degrees of freedom. In the multivariate case, the marginal distribution of is a multivariate t distribution: Suppose Then for , Proof: To prove this let and fix . Defining , observe that the PDF of the random variable evaluated at is given by times the PDF of a random variable evaluated at . Hence the PDF of evaluated at is given by : The right hand expression is the PDF for a random variable evaluated at , which completes the proof. Normal-inverse-gamma distributions form an exponential family with natural parameters , , , and and sufficient statistics , , , and . Measures difference between two distributions. See the articles on normal-gamma distribution and conjugate prior. See the articles on normal-gamma distribution and conjugate prior. Generation of random variates is straightforward: Sample from an inverse gamma distribution with parameters and Sample from a normal distribution with mean and variance The normal-gamma distribution is the same dist