There are many applications involving physical measurements which are expected to result in a probability density function (PDF) which is asymptotically Gaussian (normal) or lognormal. In the latter case, we can simply take the logs of the (positive) samples in order to obtain the former, so the math in this paper will focus exclusively on Gaussians.

For example, we would expect the distribution of radio power received at a dish to be lognormally distributed, given a sufficiently broad swath of sky to observe for a sufficiently long duration, and in the relative absence of terrestrial radio interference. However, if we were then to focus on a particular star system, the observed "experimental" PDF could substantially deviate from that "background" PDF. It might not even be lognormal if, for example, the star exhibits peaks in radio power at a few distinct frequencies.

It would therefore be useful to have a means to quantify the "surprise" factor of experimental PDFs relative to an established background PDF which is known to be, or be equivalent to, a Gaussian. If a given experimental PDF where also known to be Gaussian, then we could do this by employing the Kullback-Leibler (KL) divergence from one to the other, as Gupta appears to have done for the multidimensional case.

When the experimental PDF is not known to be Gaussian (or any PDF archetype, for that matter), the situation is more complicated, mainly because we are forced to deal with a real-valued set of samples ordered by increasing positivity -- a 1D point cloud, to be precise, although "vector" will suffice for brevity -- rather than an analytic function. Ranking the information cost of encoding such a vector, versus others arising from other experiments, under the prior assumption of the same background PDF, is the subject of this paper. We also investigate the question of ascertaining which background PDF is the most useful for the sake of discriminating anomalous from mundane experimental PDFs.