Application of POIs using formulas

[rafaellopesdesa]

Hi,

It seems that, right now, the modifiers are limited to addition of multiplication, with the POIs as normfactors being defined by simple modification POI*pdf. That's ok for most models, but some models require more complex likelihoods. The case that comes to my mind (need?) is EFT analyses which have an interference term and a squared term and, therefore, would need something like sqrt(POI)inter + POIsquare.

I don't know exactly what is the best way to implement that. I think that a simple way would be if one could define multiple POIs and if one could define the value of a modifier as a function of other modifiers. Even simple functions like polynomials would already encompass most of the case while keeping all operations very fast and efficient.

Let me know if there is anything I can do to help.

Thanks for considering.

[lukasheinrich]

hi @rafaellopesdesa

apologies for the slow response.

the way to implement this is to add a new modifier. We could call it a formulafactor, with the rule being that it can only reuse existing parameters and does not introduce new ones

{
"name": "some_name",
"type": "formulafactor", 
"data": {
  "expression" : "sqrt(x)*y", dependencies: [{"x": "mu", "y": mu2}]
}
},
{
"name": "another_name",
"type": "formulafactor", 
"data": {
  "expression" : "x**2", dependencies: [{"x": "some_name"}]
}
}

I think in the constrained language of numexpr https://github.com/pydata/numexpr and the with this constraint that no new parameters are introduced this would be quite doable.

cc @alexander-held who has also raised this issue

[rafaellopesdesa]

Hi @lukasheinrich ,

Yes, that would be great for more complex likelihood models.

I will give it a try this weekend and, if I manage to implement your idea, will come back here to report.

Thanks again!

[lukasheinrich]

I would suggest you orient yourself along the lines of the normfactor code

https://scikit-hep.org/pyhf/_modules/pyhf/modifiers/normfactor.html#normfactor

the output shape of a modifier is

(n_modifiers, n_global_samples, n_alphas, n_global_bin)

here's the meaning of the output shape

• different "formulas" to be applied
• the affect of that formula for each sample (if the sample doesn't declare/is unaffected by the formulafactor, it should just be multiplied by 1)
• the batch size of parameters.. this is probably just 1
• the global bin number (across all channels)

so if you need to compute the affect of 3 formulas and in total have 10 sample (which might or might not be affected by the formula(s) and want to evaluate the formula for 5 parameter settings and the total number of bins is 20 you would have

an output tensor of

(3,10,5,20)

[lukasheinrich]

Hi, I will likely take a stab at it since I need this for my analysis. (unless someone is already working on it?)

[alexander-held]

Hi, just to add an example of what other libraries do: uproot4 can interpret cuts / selection requirements as python (https://github.com/scikit-hep/uproot4/blob/master/uproot4/language/python.py).

I imagine that in practical applications, a very limited amount of operators will be enough for most fits: addition, subtraction, multiplication, division. Taking a power is convenient, but I guess non-integer powers are rarely needed, so multiplication is fine. Square roots are convenient too, but could be worked around by redefining normfactor->sqrt(normfactor) and then taking the power instead.

[rafaellopesdesa]

Ok, I am coming back here because I am starting a new project that could use this and this time I really want to give pyhf a try.

I would say that powers of the normfactor would already solve 99% of the cases I have in mind. The main application I have in my is cases with interference (like theta^2 * S + theta * I + B) [think EFT, for instance]

Maybe we could just have an option in normfactor that says the power which which the normfactor acts on the sample.

Would that be a simple way to implement it?

[balunas]

To bring this issue up again, I see that @lukasheinrich added some "custom modifier" feature but I can't locate any documentation on how to use this to define my own modifier. Is this something users can currently interface with? We'd like to use something like this in ATLAS di-Higgs analyses, where our POI can be a coupling which has rather complicated interference effects and can't be described by any existing modifier. To give a concrete example, we want to implement something like

yield = B + f(k)*S1 + g(k)*S2 + h(k)*S3

where k is our POI, f, g, and h are functions we define, and S1, S2, S3 are different "basis" signal distributions. Is something like this already supported in some way, or is this part of the open items remaining on this issue?

[mswiatlo]

Hi @lukasheinrich , @alexander-held ,

I wanted to bring this up again as it's starting to become a limiting factor in our analysis, and it would be really nice to avoid having to convert everything back to Roofit. Is there any documentation/testing/etc. that still needs to be done here? I also have a student who could potentially contribute to this (development, testing, etc.) if that's the limiting factor. Please let us know-- it would be really great to get this last feature that would enable a pure pyhf fit structure for our analysis!

Best wishes,
Max

[lukasheinrich]

hi @mswiatlo - let's prioritize this then.

is this the same type of formula that @balunas needs? and if so what do f(k),g(k),h(k) look like?

Importantly - is each bin modification computed only as a function of the parameters and the sample yields in that bin or are cross-bin relations important?

[mswiatlo]

Yes, we're working on the same analysis :)

[balunas]

Hi @lukasheinrich -

The ideal scenario would be if we could implement f(k) etc. ourselves in python, but in practice we're dealing with polynomials here at the moment. That is, f(k) = a + bk + ck^2 + ... There's also a k-dependent overall normalization (which gets looked up from the theory calculation). In principle that can be handled with a separate normfactor which we fix ourselves, but it would be even better if that could be included in a user-defined f(k).

There are no cross-bin relations that come in, the only dependence is on the POI.

Looking further ahead, we'd also love to use this sort of functionality with multiple POIs. I know general support for that is still in the works, but the dream would be to have something like

yield = B + f(k1, k2, k3)*S1 + g(k1, k2, k3)*S2 + ...

[lukasheinrich]

I think if the general structure is a multiplicative modifier per-sample that only depends on the parameters

sample_yield = base_sample_yield * f(POIs, NPs) is definitely workable. The summing over samples would then happen as in the other modifiers

this would not cover yet e.g. custom functions that depend on the bin location or other bins e.g. f(mass) for a mass bump or so, but seems like a good, quite general middle ground?