NP hard

After a weekend and early morning finishing the NSF proposal, I had my weekly spa on sampling and exoplanets with Goodman and Hou. For the famous Gliese 581 system, there is a fence of multiple likelihood maxima in orbital frequency (period) space, with one (that we know about) much taller than the others. The multiple optima lie on lines in frequency space (apparently), and what we want to make sure of is that the most likely is also the most probable. That is, we want to integrate the likelihood under the prior at each distinct optimum. That is, we need to compute the evidence integral for each of the optima. We would also like to find all optima, and, from among them, the best-of-all optima in both senses, but this is provably hard, so I (now, in my positivist–pragmatist phase) think that the question do we have the global optimum? is outside science. I don't think Goodman agreed with me on that, in part because if you know a lot about the likelihood function (and we do), you might be able to limit the numbers substantially. I think he is right, actually—the exponentially large number we care about might not be so large we can't tackle it—but it sure isn't easy.