#### A test for global maximum

If getting the first derivative (score function) and the second derivative (empirical Fisher information) of a (pseudo) likelihood function is feasible and checking regularity conditions is viable, a test for global maximum (Li and Jiang, JASA, 1999, Vol. 94, pp. 847-854) seems to be a useful reference for verifying the best fit solution.

I didn’t see any ways to confirm that best fit results from XSPEC or Sherpa are a global maximum without searching whole parameter space. My little understanding tells that many fitting algorithms do not guarantee a global maximum. By checking that the best fit solution is the global maximum and subsequently, the obtained error bar is expected to have the nominal coverage, we could save efforts of searching whole parameter space.

1. ##### aneta:

The global maximum is only a part of the issue.

The global maximum could be located in “unphysical” part of parameter the space. Do these test work in such situation, i.e. when the parameter space is constrained within the physically acceptable solutions while the global max is not?

When you work with observations you typically think about how well the global max is constrained given the data. Within the “confidence” regions in the parameter space you can find a range of acceptable solutions, so even if you determine that yes, this is a global max, your goal is to collect such data that this range is small. so if within the constrained 90% intervals you have many “local” maxima does it really matter that you landed in one of them? I think you need to be sure that you can constrain the 90% intervals well.

07-02-2008, 8:14 pm
2. ##### hlee:

Very good point. I do understand that and that’s the reason I wrote my first aas: normalization, but not from an astrophysicist’s viewpoint. No statistic is panacea, I believe. By following generalized information criterion notation, J (first derivative) and I (2nd derivative) are what we need for this test which does not promise finding a global maximum. It only tells that if a best fit satisfies J^2+I~0, it is a global maximum. For a better answer to your question, I’d rather quote a few sentences from the paper:

… if the test rejects, then one concludes that there is either a problem of model misspecification or a problem of a best not being a global maximum, but one cannot distinguish between these two. However, in many cases other techniques are available for model checking. … to the best of our knowledge no statistical method is available for checking whether a given root corresponds to a global maximum, even if the the model has been determined correct. This is the main reason for writing this article.

Thanks to physics, astrophysicists know evidently the chosen model is correct but the model could have a complex parameter space providing multiple solutions. I thought this test for a global maximum would be useful under this circumstance. It’s cheaper to run the fitting process with different initial values instead of searching full parameter space. On the other hand, the circumstance that the global maximum in unphysical part of the parameter space does not satisfy the regularity conditions.

07-02-2008, 11:28 pm