Reviewing binned Compensator Based Inference

The results described in this paper are useful and significant. One reservation I have is that this draft is very close to the previously published "Banerjee and Algeri (2026)". The submitted draft extends that method — derived for independent and identically distributed data — to binned histograms with Poisson counts. The conceptual contribution is largely inherited from the published work, and, as I explain below, the extension also weakens the main motivation for the method. So its not clear to me if this warrants a separate standalone publication. I have one substantive critique specific to the binned case, followed by minor comments.

The original paper shows that the effect of a misspecified background on the signal hypothesis test is captured by a single scalar — the compensator — equal to the component of the background misspecification aligned with the signal. When a background-only dataset is available the compensator is identifiable, and a test for the presence of signal can be constructed that controls the type I error rate for essentially any choice of the postulated background, however poor, subject only to mild support/integrability conditions.

The submitted draft shows that the same can be done with binned datasets.

My concern is that, in the binned case, the advantage of the compensator is unclear. Section 3 assumes a binned background-only histogram, which is what identifies the compensator. But a binned histogram is already a nonparametric description of the background: unlike the iid case, there is no arbitrary functional form to impose on f_b, and hence none of the shape-misspecification bias that the compensator is designed to absorb. If binned f_b and f_s are available, η in Eq. (1) can be estimated directly by a binned maximum-likelihood fit, with the presence of signal assessed by the profile likelihood-ratio test. With a correctly specified binned background the model is correctly specified, the standard boundary asymptotics (½χ²₀ + ½χ²₁) apply, and there is no misspecification to compensate — the protective role of the compensator, which is the entire point of the iid construction, is unnecessary precisely in the regime where f_b is directly available.

Of course the background-only histogram is a finite sample of f_b rather than the truth, so a fair comparison is against a conventional binned fit that propagates the control-sample statistics in the usual way. Both approaches use the same finite control histogram; the question is which uses it more efficiently. The manuscript does not address this. I would ask the authors to (i) state explicitly what advantage the compensator provides over a standard binned template fit in the setting of Section 3, and (ii) support the claim with a direct comparison, especially in the relevant low-count-per-bin regime. With out this, it is not clear that the compensator buys anything once the data are binned.

This critique does not apply to Section 4, where no background-only sample is available and f_b cannot be used directly. However, as discussed above, Section 4 closely parallels the corresponding treatment in the already-published iid paper.

Other than that I am happy with the submitted draft. A few minor comments below.