It is not a widely discussed subject , and the concept was certainly not used in my time ( twenty years ago) . I found this definition:
When using pseudo-data, generated with the help of Monte Carlo simulations, the truth distribution $x^{truth}$ is known, so the unfolding result $ˆx$ may be directly compared to it. Such comparisons, where pseudo-data are unfolded and compared to the truth are often called closure tests.
So I understand it is a comparison of theoretically known distributions, which are used to generate the "unfolding" result, and it should be within the estimated errors for the pseudo data .
This is a use in high energy data:
Closure Test for Cross Section Analysis
Closure means that the acceptance and efficiency corrections work
It is still too esoteric for me, lets hope that somebody who has used the method replies .
Also, is using the same MC for analysing and then for final comparison a good idea?
Monte Carlo is a method of integration. If one can do the integration analytically , one can compare with the simulated data which has statistical and systematic errors, and these should agree within errors. It is two different comparisons of monte carlo data. The objective is to catch and discrepancies within the integration, so that new physics would appear in comparison with data.