Here is a beautiful ill-posed statistics problem in astronomy that comes up over and over again: Image A in bandpass B with point-spread function C overlaps image D in bandpass E with point-spread function F. Which sources in image A match which sources in image D? When the bandpasses and point-spread functions become arbitrarily different (for example, when you compare an HST image to a SCUBA image), the problem becomes hard.
Not just hard, but seriously ill-posed. My position is that all important questions in science are ill-posed questions. Roweis and I spent some time discussing how to convert this problem into a well-posed statistics problem and then how to solve that well-posed problem. This is on the critical path towards building a generative model of every astronomical image ever taken.
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