Backward message passing in variational Bayesian inference I have come across in a research paper that,


I do understand the logic. But the paper has't mentioned about the way of updating $\eta_{t}$. When I asked from the authors they said when we equate the first derivative to be zero, we could find how to update $\eta_{t}$.
But, I couldn't exactly figure out how to relate $\eta_{t}$ and $\eta_{t+1}$ from the first derivative.
Any insight about this is greatly appreciated.
Thanks.
Note: 

and

 A: Because of the nonlinear function $\pi(z_{t+1})$, you cannot directly solve for the joint mode.  You would have to apply "iterative methods, e.g. Newton's method" as done for computing $\alpha$ earlier, with details described in (the non-existent) Appendix A.
I have a hard time believing that the methods in this paper are going to give good approximations.  The reason is that when they solve for the joint mode they aren't using all of the information available about the variables.  During the $\alpha$ computation they find the joint mode without using backward information.  During the $\beta$ computation, they find the joint mode again but without using forward information.  Only when computing posterior marginal do they compute the joint mode using all of the information.  The approximation they are making is very local, so if the mode computed using $\alpha$ is far away from the mode computed using both, then $\alpha$ will be accurate in a location that you don't care about, and inaccurate in the location that you do care about.
