THE JACOBIAN OF THE MATRIX EXPONENTIAL TRANSFORMATION: BAYESIAN INFERENCE FOR A COVARIANCE MATRIX

                        

                                                             Senior co-author  Jan Magnus

                                                        Let       C= exp (A)

      where C is a positive definite covariance matrix and A=log C denotes the matrix logarithm of C

             Then

                                Magnus, Pils, and Centana(2021)

           were the first to complete the monumental task of deriving a closed form expression for the Jacobian of this transformation. My congratulations to the authors!!

            For example,

            (1) If C possesses an inverted Wishart distribution, then the density of A may be stated in closed form.

            (2) If the upper triangular elements of A possess a matrix normal distribution, then the density of C cam be stated in closed form.

            This result will find many applications in Bayesian Inference for a Covariance Matrix. The large previous literature includes:

                                     Leonard and Hsu (1992)

                                     Lesage and Pace (2002)

                                    Hsu, Sinay, and Hsu (2012)

                                     Deng and Tsui (2012)

                                     Asai and McAteer (2020)