Can I split this likelihood term

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.

• You can if the residuals are independent across spatial dimensions. Otherwise, if you know the covariance structure (somehow that doesn't seem likely) you can apply a transform to the length-3 vector of a residual that transforms it to a length-3 vector of independent Gaussian variates, and you can split the resulting likelihood as you'd like. – jbowman Feb 19 '14 at 18:54
• What are $w$ and $z$ here? Is $t(\cdot)$ a known function? What transformation are you talking about that generates index $d$? Can you give some context to the problem? Is this a predictive model for air pollution? Inference on fMRI data? – AdamO Feb 19 '14 at 18:56
• Sorry for the lack of information. $w$ are the transformation parameters that parameterise the transformation $t$. $t(.)$ is a known transformation. I figured out that it does not need to be written like that to apply EP and it probably could not be written across the dimensions because as bowman pointed out the residuals are not independent across dimensions. – Luca Feb 19 '14 at 20:51