I am reading A Tutorial on Graph-Based SLAM.Grisetti, Kummerle, Stachniss & Burgard
On page 5, the error function is introduced as follows
$$e_{ij}(x_i, x_j) = z_{ij} - \hat{z}_{ij}(xi, xj)$$
here $z_{ij}$ is the mean of virtual measurement and $\hat{z}_{ij}(x_i, x_j)$ is the prediction of the virtual measurement. The following image supplements the description
The Algorithm 1 (on page 6), requires $e_{ij}$ as input. My doubt is regarding the calculation of $e_{ij}$. I need both $z_{ij}$ and $\hat{z}_{ij}$ to calculate $e_{ij}$
The evaluation of $\hat{z}_{ij}$ is dependent on the robot poses $x_i$ and $x_j$.
In turn, these robot poses $x_i, x_j$ (as well as $z_{ij}$) are calculated using $z_{raw}$ (incrementally with Odometry?) and to calculate $\hat{z}_{ij}$ we are again going to (indirectly) use $z_{raw}$. And that does not make sense because then $e_{ij}=0$?
Surely, I'm missing something about how $z_{ij}$ and $\hat{z}_{ij}$ differ!
Kindly help me resolve the above doubt! Any concrete example of $z_{ij}$ and $\hat{z}_{ij}$ are appreciated.