Exclusion limits on particle dark matter

I'm trying to make sense of how the upper limits of the cross-section of particle dark matter (WIMPs) are computed, like in the figure below. This is taken from an article on dark matter searches. Quoting from the article:

The number of selected events in the region of interest is used to set a limit on the dark matter cross-section. [p.2]

and then

A limit is set by assuming all events are originating from the dark matter scatters and the maximal possible cross-section is determined by using Yellin's optimal method, which is considered to provide a conservative limit. [p.3]

In skimming this article about the statistical methods used in particle dark matter searches, the optimum limit method is given by eq. 13 on page 15

$$x_i = \int_{E_i}^{E_{i+1}} \frac{dN}{dE}(\sigma_{\text{scatt}} )dE \ .$$

I'm not sure if this is the expression that I'm looking for.

In its treatment, the concept of a confidence interval is used, which still eludes me.

Furthermore, I've read on CLs methods which according to Wikipedia is

a statistical method for setting upper limits (also called exclusion limits) on model parameters, a particular form of interval estimation used for parameters that can take only non-negative values.

Mind you, I have not taken a rigorous course in probability theory, so a brief exposition in how the concept of the confidence interval is involved in calculating the exclusion limits, in the figure above, would be highly appreciated. :)

• Like so many questions about the analysis of significant experiments this can only be condensed into a reasonable size answer by assuming that the reader is already fairly familiar with the field. For instance, any discussion you might find in a paper would expect you to know enough about confidence limits to interpret them even if you aren't familiar with the particular method used to set them in this paper. – dmckee Jun 5 '18 at 0:30
• Well, I'd tolerate a fair bit of handwaiving in this case. :) – Mussé Redi Jun 5 '18 at 7:15

• I think I'm way over my head with this question, as I've not done the proper research. I don't know how to compute confidence intervals;how the cross section is inferred; what you mean when you say we should discard all ($m$,$\sigma$) pairs less then a certain upper limit. – Mussé Redi Jun 24 at 12:14