Acceptance is not properly propagated during MapDatasetOnOff stacking about gammapy HOT 1 CLOSED

The issue most likely comes from the way we compute stacked acceptances.

Reminder, for excess calculation with ON-OFF data:

• for a single dataset we have $N_{excess}$ and $\alpha$:
  $$N_{excess} = N_{ON} - \alpha N_{OFF} $$
• with $\alpha = \frac{Acc_{ON}}{Acc_{OFF}}$ the ratio of ON and OFF acceptances
• to guarantee correct number of excess when stacking one has to compute an $N_{OFF}$-weighted $\alpha$, such that:

$$ \sum_i N_{excess,i} = \sum_i N_{ON,i} - \bar{\alpha} \sum_i{N_{OFF,i}}$$

• with:
  $$\bar{\alpha} = \frac{\sum_i \alpha_i N_{OFF,i}}{\sum_i N_{OFF,i}}$$

Usage in Gammapy

• This principle is applied :
  • when stacking ON-OFF datasets
  • when computing the correlated maps in ExcessMapEstimator

A caveat

• this does not say what to do with the ON and OFF acceptances, in Gammapy we assume that the stacked acceptances are:
  $$\bar{A}cc_{ON} = 1 \quad \text and \quad \bar{A}cc_{OFF} = \frac{1}{\bar{\alpha}}$$
• This choice is problematic as it removes all information on where the events actually come from
• We should instead stack $Acc$ to represent the sum of acceptances taking into account the safe mask:

$$ \bar{A}cc_{ON} = \sum_i Acc_{i} \times \epsilon_i $$

• and
  $$\bar{A}cc_{OFF} = \sum_i Acc_{i}\ \epsilon_i \times \frac{\sum_i N_{OFF,i}}{\sum_i \alpha_i N_{OFF,i}}$$

from gammapy.