One of the classic examples of this is three interpretations of "randomly select a point from a circle."

In a separate comment, Kurros worries about cases with "no preferred parameterisation of the problem". I have the same worry as both of you, I think. I guess I'm less optimistic about the resolution. The parameterization seems like an empirical rabbit that Jaynes and other descendants of the Principle of Insufficient Reason are trying to pull out of an a priori hat. (See also Seidenfeld <pdf> section 3 on re-partitioning the sample space.)

I'd appreciate it if someone could assuage - or aggravate - this concern. Preferably without presuming quite as much probability and statistics knowledge as Seidenfeld does (that one went somewhat over my head, toward the end).