[anon946]

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.

[lupire]

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.