@Hari The use of squared errors, or root mean square, is a common way of creating a useful measure of quality, when otherwise ignorant of how to weight errors due to other a priori knowledge. It's not handed from god, though. There are other ways if, for example, there exists some correlation between one error source and another or if the distribution shapes are far from Gaussian. Not sure what you are questioning, though. The log equation itself is unassailable math. So it must be something else. In general, I find their approach useful enough. What do you want from this discussion?
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