The essence of my question is this:
Let $Y \in \mathbb{R}^n$ be a multivariate normal random variable with mean $\mu$ and covariance matrix $\Sigma$. Let $Z := \log(Y)$, i.e. $Z_i = \log(Y_i), i \in \{1,\ldots,n\}$. How do I compare the AIC of a model fit to observed realizations of $Y$ versus a model fit to observed realizations of $Z$?
My initial and slightly longer question:
Let $Y \sim \mathcal{N}(\mu,\Sigma)$ be a multivariate normal random variable. If I want to compare a model fit to $Y$ versus a model fit to $\log(Y)$, I could look at their log-likelihoods. However, since these models aren't nested, I can't compare the log-likelihoods (and stuff like AIC, etc.) directly, but I have to transform them.
I know that if $X_1,\ldots,X_n$ are random variables with joint pdf $g(x_1,\ldots,x_n)$ and if $Y_i = t_i(X_1,\ldots,X_n)$ for one-to-one transformations $t_i$ and $i \in \{1,\ldots,n\}$, then the pdf of $Y_1,\ldots,Y_n$ is given by $$f(y_1,\ldots,y_n)=g(t_1^{-1}(y),\ldots,t_n^{-1}(y))\det(J)$$ where $J$ is the Jacobian associated with the transformation.
Do I simply have to use the transformation rule to compare
$$l(Y) = \log(\prod_{i=1}^{n}\phi(y_i;\mu,\Sigma))$$ to $$l(\log(Y))=\log(\prod_{i=1}^{n}\phi(\log(y_i);\mu,\Sigma))$$
or is there something else I can do?
[edit] Forgot to place logarithms in the last two expressions.