Self-contained proof for Wishart distribution

[yousuketakada]

Show the normalization (B.79) as well as the expectations (B.80) and (B.81) of the Wishart distribution (B.78) in a self-contained manner.
We have already pointed out that some appropriate citation, e.g., Anderson (2003), is needed for the Wishart distribution because it has been introduced without any proof. However, most multivariate statistics textbooks, including Anderson (2003), motivate the Wishart distribution differently from PRML; they typically introduce the Wishart distribution as the distribution over the scatter matrix.
The derivation of the Wishart distribution along this line is indirect for our purpose (we are mainly interested in its conjugacy). I would rather like to show the normalization (B.79) as well as the expectations (B.80) and (B.81) directly just as we have done for the gamma distribution (2.146).
We show the Wishart distribution based on the matrix factorization called the Cholesky decomposition as well as the associated Jacobian. We also introduce the multivariate gamma function, which simplifies the form of the normalization constant (B.79). The expectations (B.80) and (B.81) are shown by making use of the fact that the expectation of the score function vanishes.