# Log Expectation of Inflated Determinant of Wishart Distribution

Let $\Lambda \sim \mathcal W(\nu, \Psi)$, i.e., following a $n \times n$ dimensional Wishart distribution with mean $\nu \Psi$ and degrees of freedom $\nu$. The expectation of the log determinant of $\Lambda$ is $$E(\log|\Lambda|) = n \log 2 + \log |\Psi| + \sum_{i = 1} ^ n \psi\left(\frac{\nu - i + 1} 2\right),$$ where $|\cdot|$ denotes the determinant of a matrix and $\psi(\cdot)$ denotes the digamma function $\frac d {dx} \log \Gamma(x)$.

Now, I would like to compute the expectation of log determinant inflated by a positive definite matrix: $$E(\log |\Lambda + \Sigma |),$$ where $\Sigma$ is the positive definite matrix to inflate the determinant.