User:Lkbwx/sandbox


 * $$\begin{align}

\operatorname{H}[\mathbf{X}] &= \tfrac{n}{2}\ln|\mathbf{V}| +\tfrac{np}{2}\ln(2) + \ln\left (\Gamma_p(\tfrac{n}{2}) \right ) -\tfrac{1}{2}(n-p-1) \operatorname{E}[\ln|\mathbf{X}|] + \tfrac{np}{2} \\ &= \tfrac{n}{2}\ln|\mathbf{V}| +\tfrac{np}{2}\ln(2) + \tfrac{1}{4} p(p-1) \ln(\pi) + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n}{2}+\tfrac{1-i}{2}\right ) \right ) \\ &\qquad \qquad -\tfrac{1}{2}(n-p-1)\left(\sum_{i=1}^p \psi\left(\tfrac{1}{2}(n+1-i)\right) + p\ln(2) + \ln|\mathbf{V}|\right) + \tfrac{np}{2} \\ &= \tfrac{n}{2}\ln|\mathbf{V}| +\tfrac{np}{2}\ln(2) + \tfrac{1}{4} p(p-1) \ln(\pi) + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n}{2}+\tfrac{1-i}{2}\right ) \right ) \\ &\qquad \qquad - \left ( \tfrac{1}{2}(n-p-1)\sum_{i=1}^p \psi\left(\tfrac{1}{2}(n+1-i)\right) + \tfrac{1}{2}(n-p-1)p\ln(2) + \tfrac{1}{2}(n-p-1)\ln|\mathbf{V}|\right) + \tfrac{np}{2} \\ &= \tfrac{p+1}{2}\ln|\mathbf{V}| +\tfrac{1}{2}p(p+1)\ln(2) + \tfrac{1}{4}p(p-1) \ln(\pi) + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n}{2}+\tfrac{1-i}{2}\right ) \right ) -\tfrac{1}{2}(n-p-1)\sum_{i=1}^p \psi\left(\tfrac{1}{2}(n+1-i)\right) + \tfrac{np}{2} \end{align}$$