Man, that sleeping beauty digression at the end. I'm so easily aggroed.
I wonder if I should write up a rant or a better attempt at exposition, against this "clearly the problem is underdetermined" position. The correct answer is somewhat difficult and all expositions I've seen (or written!) so far have had big shortcomings. But even if one didn't know the right answer, one should readily conclude that a principled answer exists. We assign things probabilities for fairly deep reasons, reasons undisturbed by trifles like the existence of an amnesia pill.
That aside, good talk :)
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Just because it is obvious to you doesn't mean that everybody immediately jumps to it. Me included and I like the statistics classes back then. Could you please point to the wheels?
The wheels in this case come from robust statistics.
One example of a good robust estimator for the center is the [truncated mean]https://en.wikipedia.org/wiki/Truncated_mean). To put it simply: throw away the lowest x% and the highest x% of the samples, and take the mean of the rest. If x=0 you get the regular mean, if x=50% then you get the median.