For the discrete case, it seems that a better thing to do is consider the likelihood of getting that number of heads, rather than the likelihood of getting that exact sequence.
I am not sure how to handle the continuous case, however.
Of course you ignore irrelevant ordering of data points. That's not the issue.
The issue, for discrete or continuous (which are mathematically approximations of each other), is that the value at a point is less important than the integral over a range. That's why standard deviation is useful. The argmax is a convenient average over a weightable range of values. The larger your range, the greater the likelihood that the "truth" is in that range.
If you only need to be correct up to 1% tolerance, the likelihood of a range of values that have $SAMPLING_PRECISION tolerance is not importance. Only the argmax is, to give you a center of the range.
lupire|2 years ago
The issue, for discrete or continuous (which are mathematically approximations of each other), is that the value at a point is less important than the integral over a range. That's why standard deviation is useful. The argmax is a convenient average over a weightable range of values. The larger your range, the greater the likelihood that the "truth" is in that range.
If you only need to be correct up to 1% tolerance, the likelihood of a range of values that have $SAMPLING_PRECISION tolerance is not importance. Only the argmax is, to give you a center of the range.