Consider a reparameterised Gaussian function, $x = \mu + \sigma \cdot n$, where $\mu \in \mathbb{R}^D$, $\sigma \in \mathbb{R}^{D \times D}$ and $n \sim \mathcal{N}(0,\mathbb{1})$ is a random vector sampled from the multivariate normal distribution. If I bound the output of my function by applying $y = \tanh x$, what is the log-probability density $\log \rho(y)$ in terms of $\mu, \sigma, n$? Here, the $\tanh(\cdot)$ function is applied element-wise on the vector $x$.
If we used a standard multivariate normal, then this question would be simpler, however my use case required the reparametrisation trick. Places checked so far: Appendix C of https://arxiv.org/pdf/1812.05905.pdf (SAC paper) shows how to handle the passing of the $\tanh(\cdot)$ function using a Jacobian (standard change of variables), but does not specify the form of the Gaussian log-probability density function for $x = \mu + \sigma \cdot n$, so a simpler question could also be what is the log-probability $\log \rho(x) $?
