In forward path tracing method, we don't know where the light comes from, therefore, we sample them to find them. However, light comes from light sources. Therefore, it is natural to distribute light from the light source and then sample the environment to find where is bright. This could be more efficient. How this can be done without biasing is an issue and the paper described it.
There are some cases that forward path tracer hardly handles some paths. For example, a projector projects an image to a wall. Forward path tracer creates paths from a view point and hits the wall, then samples the lens of the projector. Just looking up the light source (the projector) can be done, but, it is difficult to see the image on the wall since such a dense samples must be on the lens without a direct guidance. It's not impossible for forward path tracer, but practically it is hard and mostly we could not see an image.
In this case, the projector projects an image to the wall and sample the wall would be more efficient. This is my understanding of the basic idea of bidirectional path tracing.
Overview idea of Metropolis sampling algorithm
There is a case that bidirectional path tracer also has problems to find some kind of paths. Veach's paper's example is that: ``there are two rooms, one room has a light source, other is dark room that doesn't have any light source. There is a door in-between the rooms, that is almost closed. The camera is in the dark room.'' The light only passes through the gap of the door. To avoid bias, we could not sample the door's gap only. (Is we do that, then there will be a path that can not be detected.) If we can find a nice path once, we should exploit something similar path, but not the same ones. This is the Metropolis method. (Metropolis is a person's name who proposed this method.)
Veach proposed the initial guess is done by a bidirectional path tracer. If we could find a path that can reach to any light source, we perturbate the path and exploit more paths exist or not.
By the way, I used some kind of variational method for geometry processing. A problem is interpreted to some kind of potential energy problem and is applied a variational method. But the energy design is the problem. Once you have a nice energy, then perturbate your solution and get the better solution. For the light transport problem, it seems difficult to find a derivative, but, the basic idea of Metropolis algorithm looks surprisingly similar.
I enjoyed the Veach's paper. This paper uses a kind of basic and orthodox methods. Basic and orthodox are well established and so many times challenged, therefore I like them.
My personal annotations of Veach's paper ends now. Thanks a lot. So, what will be the next?