I have a likelihood that is modelled using the IID distributed noise assumption. Now the likelihood at a 3D location $i$ is normally distributed with 0 mean and some precision $\sigma$. So, I can write it as a product:

$$ L(E) \propto \prod_{i} \exp^{-0.5 * e_i \sigma e_i} $$

Now each location $i$ is a location in 3D and the residual term $e_i$ is obtained by the quantity $y - t(x, w)$ where $y$ and $x$ are observed and $t$ is a non-linear transformation at each location $i$.

Now, my question is can I split the likelihood even further along the 3 spatial dimensions... i.e.

$$ L(E) \propto \prod_{i} \prod_{d} \exp^{-0.5 * e_{id} \sigma e_{id}} $$

where now $e_{id}$ is the residual now generated by applying the transformation along the spatial dimension $d$. Since the noise is modelled at each 3D location, I wonder if this can be done as the noise level along the $x$, $y$ and $z$ dimension for a location $i$ is the same and hence perfectly correlated.

The reason I ask is that my inference scheme using Expectation Propagation gets a lot simpler if I can split this likelihood terms into univariate Gaussians.