From: Frank Masci Subject: upper limits Date: October 21, 2011 3:31:53 PM PDT I went to bed last night thinking about upper limits in the context of source detection in astronomical images. It was a mistake because I tossed and turned while trying to get the logic right. I don't see a consistent formalism in the literature and therefore would like to develop something that's foolproof, raises no arguments(!), and can be quantified probabilistically. At the end of X's WHPhot SIS [http://wise2.ipac.caltech.edu/docs/doc_tree/sis/pht01], I noticed: threshold = 2 * RMS if ( f < threshold) then if (f < 0) then upper_limit = threshold else upper_limit = f + threshold This doesn't feel right to me, although the definition can get philosophical. I don't think an upper limit should be quantified in terms of the high end of some confidence interval about a measured possible noise fluctuation (f) as written above. The actual photometric detection s/n limit (under a null distribution [H0] of zero-mean random noise about the background) should drive the definition of an upper limit (I think!). Even though the above gives a flux bound that's probably conservative, I don't know whether it's optimal and I'm not sure how to assign a probability. I see it as either black or white: either you detect something and have something certain to say about the measurement and its uncertainty, or you don't and are way less certain of what it means. For example, a fluctuation can be either above s/n = 3, or below. Suppose one picks s/n=3 as their threshold. If s/n > 3, then the (one-tailed) probability that it's a spurious noise spike under a Gaussian H0 is ~0.135%. One can then reject H0 for that measurement and declare it to be a "real source", i.e., a systematic excursion in the noise distribution where one now is reasonably confident that its true flux is > 0. They can then assign a +/- n*sigma confidence interval to its measured flux, f. This interval just means there's an x% probability the true flux lies within f +/- n*sigma. I.e., repeated independent observations of the purported source, giving flux measurements f_i (and assuming no intrinsic variation), means x% of the intervals: f_i +/- n*sigma_i with contain the truth (as N --> infinity of course). However, if a single measurement has s/n < 3 (still under a H0 of pure random noise), say s/n=2.9, then obviously it doesn't satisfy our maximum tolerable fraction of false positives at this cut (here <~0.00135 for Gaussian-noise). Without any prior knowledge of what's there, this sub-significant measurement should be treated as an ordinary noise excursion, not a candidate for a possible systematic excursion (i.e., a real source that may hover around s/n~2.9 on repeated observation). For a single realization (measurement), I would say an upper limit on its flux is loosely < 3*sigma, or more conservatively < 5*sigma. One can then assign a corresponding (high-tail) probability that its true flux is less than this limit under H0 (and of course assuming no intrinsic variability). One can declare this to be a real source by optimally combining (independent) repeated observations and beating down the noise, so it can be made to exceed the s/n threshold. The observer can now make a more judicious decision, with confidence estimated from a new p.d.f. describing the real (f_true>0) flux fluctuations from this source, instead of the H0 noise-distribution used for the initial detection process. So, my claim is that upper limits should be defined from the initial detection process alone, not a confidence interval based on the assumption that any positive fluctuation below some user-defined threshold is a real persisting source (i.e., a systematic excursion with noise p.d.f. centered around some f_true>0, instead of f_true=0 for the initial detection H0). A fluctuation shall remain a noise-fluctuation under some H0 unless it exceeds some judicious threshold and is declared a source. Of course, then an upper limit is no longer needed unless the threshold is redefined (increased). The best one can do when below "the threshold" is assume it came from a noise distribution with f_true=0. You probably think it's not a big deal and there's no problem, but I would appreciate your thoughts on this. Regards, Frank