NormalInverseWishart

The Normal-Inverse-Wishart distribution is a conjugate prior for the parameters of a multivariate normal distribution. It is commonly used in Bayesian statistics for its mathematical tractability and the ease with which it can be updated with new data. The distribution is defined by four parameters: a location parameter (mu), a scale matrix (Lambda), a degrees of freedom parameter (nu), and a scale matrix (Psi). The location parameter (mu) represents the mean of the distribution, while the scale matrix (Lambda) and (Psi) are positive definite matrices that control the spread and shape of the distribution. The degrees of freedom parameter (nu) is a positive real number that influences the distribution's tail behavior. The Normal-Inverse-Wishart distribution is particularly useful in Bayesian inference for multivariate normal data, as it allows for the incorporation of prior information and the updating of beliefs as new data becomes available. The posterior distribution of the parameters of a multivariate normal distribution, given a Normal-Inverse-Wishart prior and some observed data, is also a Normal-Inverse-Wishart distribution. This conjugacy property simplifies the Bayesian updating process and makes the Normal-Inverse-Wishart distribution a popular choice for prior distributions in Bayesian analysis of multivariate normal data.