The Naive question of Profile-Likelihood-Test-Statistic

Roofit and RooStats
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Dear experts.
I try to understand the Invert hypotest by using the Profile-likelihood-test-static. I have below questions about your RooStats

  1. For example, for a simple Poisson model Pos(n|s+b), where the null hypothesis (H0) assumes the S+B model and the alternative hypothesis (H1) assumes the B model, when using RooStats to obtain the distribution of likelihood ratio test statistics, I would like to ask: when RooStats generates Toys, does it sample from the distribution Pos(n|s+b) under the S+B model or from the distribution Pos(n|b) under the B model, in order to obtain the distribution of the test statistic under different hypotheses? According to my testing, I found that the data are generated according to the S+B model assumption, which is confusing to me. Why is it not generating toy MC under the H1 hypothesis to obtain the distribution of the likelihood ratio test statistic under different hypotheses?
  2. another things is that the name of this test statics is misleading, because I find it is the -log(profile likelihood ratio).
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I guess @jonas can help you.

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