Equation 9.8's weight is quite similar to the weight used in my favorite operator, Laplacian operator. Here Veach said this is heuristic, but I see this condition is carefully chosen. There are positivity and partition of unity. Following these conditions, we can guarantee the solution's sup and inf can not be out of range of the value of the sampling values. It might be misleading, but, briefly speaking, this operator is an extension of averaging. One amazing property of average is the average never over the maximal value and never less than the minimal value. For example, the average of 5 and 10 is in-between 5 and 10, never larger than 10, never less than 10. You might hear I said something so simple in too complicated way. But, this property is still valid when these weights are applied in here. (9.8's weight gives us Affine combination of the samples.) If you think about the weights, you might need to think that these property can be kept or not. This is kind of natural that when we combine several solutions, this is great of this paper.
p.267 Balance heuristic 2
Figure 9.3 is the pseudo code of balance heuristic estimator.
I would like to add some annotation of this pseudo code since I think this code is important.
- i means i-th sampling technique. k is also used as k-th sampling technique, but this is used to distinguish different iteration.
- n is number of sampling techniques. Therefore, i = 1, 2, ..., n. k is also the same.
- n_i is the number of samples of i-th sampling techniques. Therefore, the line 2's N is total number of samples. For example, technique 1 samples 5, technique 2 samples 3, then n_1 = 5, n_2 = 3, N = 8.
- p_i is the probability function of i-th sampling technique.
- X is a sampling point.
As usual, thanks to Leo.