I have a theoretical question about optimal similarity metrics for comparing data sets.
In reading this linked paper, pp. 488-489 , I read the following 2 interesting statements. On page 488:
In the ... scenario in which $A$ and $B$ differ only by Gaussian noise, then it can be shown that $\rm SSD$ [Sum of Squares Difference] is the optimum measure
And on page 489:
If the intensities in images $A$ and $B$ are linearly related, then the correlation coefficient $\rm CC$ can be shown to be the ideal similarity measure
Are there are other conditions under which other similarity metrics [e.g. Mutual Information] can be proven to be optimal, or under which the above 2 metrics ($\rm SSD$ and $\rm CC$) are also optimal? The linked PDF focuses on image registration. I am thinking mostly in terms of image registration/image fusion similarity metrics, but would also be interested in more general similarity between any 2 or more datasets (don't have to be images).
 P. A. Viola, Alignment by maximization of mutual information. Ph.D. thesis, Massachusetts Institute of Technology, 1995.