Combination of different nuisance parameters in HistFactory


i was trying to find a resource that describes how pulls of different nuisance parameters are combined.

Say i am just looking at one bin with nominally 100 events. Now i have two nuisance parameters.
At their +1 uncertainty the first one predicts 105 from the shape component and 110 from the normalization effect.

With the other predicting 98 and 108 respectively.

If both of them are set to their +1 level, what would the final number of events be?

I have found this equation

in a thesis that uses HistFactory. This would indicate that first all shape effects are summed and then all normalization effects are multiplied. Is this the case, and if so is there a citeable reference?



If your nuisance parameters are all at +1 sigma value of their uncertainty, the expected value on the number of events will be the one maximising the likelihood. See the HistFactory note for the detail of the equations describing the likelihood (e.g. equation 5 and 6).



Thanks for your reply. If I understand correctly then the basic approach is to calculate the expected number of events for a particular nuisance parameter combination and then get the likelihood for this combination. Then adapt the parameters so that this likelihood is maximized.

And my question goes into how this expected number of events is calculated when multiple parameters are pulled.

I think I now found that my question is also addressed in the interpolation section. There it seems that for the linear Interpolation different NPs are added together, where as for the exponential they are multiplied. Equations 14 and 15 vs 17 and 18.

Now the question would be how this looks for the current defaults?

In the note it seems to say that for the overallsys they are multiplied for option 4. Is that still the case for them in the current version (or rather was in 6.08?). And how does it look for shape sys? As far as I understand there is now also a polynomial+linear inter/Extrapolation for that now, which did not exist in the note.

@moneta Do you have more information on this or could you point me to the relevant parts of the code? It seems to me that the combination of different parameters depends on the interpolation setting and the note is missing one of the current default interpolations. The current default interpolation for OverallSys should be the (Poly+Exponential) which gets combined as the product. Is that easy to verify in the code and also check how that works for the (Polynomial+Linear) case which is the current HistoSys default inter/extrapolation?

@moneta Could you point me to a resource or code that shows how the different NPs are combined for the current default interpolations? It seems interpolation dependent and one of the two defaults did not exist when the note was created.

When all nuisance parameters are situated at the +1 sigma value of their uncertainty, the anticipated number of events will correspond to the value that maximizes the likelihood. Refer to HistFactory note 4 for a detailed explanation of the equations that describe the likelihood, such as equations 5 and 6.

Hi @jokker89 , thanks for the reply.

The equation of intererst for me here should be equation (6), right?

And then the individual components depend on the chosen interpolation scheme, right?

For example equation 14 shows that, for linear interpolation, different uncertaintiy components are summed…

Meanwhile equation 17 shows that for exponential interpolation they are multiplied.

The current defaults are polynomial interpolation with exponential or linear extrapolation.

Equatio 24 shows that for the exponential extrapolation the combination uses the product. However i do not know how it works for the linear extrapolation?

Is it also fine to assume that these values there still hold in current releases? Or is there someplace in the code that i can look to verify these?

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