It will take me some work to completely understand your mathematics, So I'll make some qualitative comments here.
Go game are built with exact information. However, due to its complexity, statistical description becomes not only meaningful, but also an important tool to understand the game. For example, the concepts, such as influence and thickness are all of much statistical sense. On the other hand Go game is not yet a statistical ensemble. This is where the difficulties are. There is almost no existing theory that deal with this transition from the complete information to a statistical ensemble. Probably the 2, 3,...N-body correlation functions are as close as we have so far.
To point out the deficiency of a statistical model, I'll comment on the example given in the paper.
The first example, consider the balck stone at C17. If black play B17, he will make some territory there. However, if white play at C16, C17 will be captured. Thus, the territory at that area depends on who plays first. Without specifying who plays first, the territory there probably should be the average between the white and black territories if either one play there first.
Second example, consider the position at N3. If balck plays at N3 first, it worths gote 3 points for black. However, if white play there it's sente which black cannot ignore. Thus, the position N3 should be more white than black.
The black moyo at upper left is also a good case for study. However, I don't have any good point on this subject. Somebody else may have some good idea.
Daniel Liu
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