docs: learn nb on covariance matrices

[lukasheinrich]

Description

spent some more time with MINUIT code to see how it does the covariance calculation and putting the lessons into a notebook. What I was initially confused by is how it recomputes the full hessian only by transforming the hessian in the internal coordinates. Turns out it uses a property that only holds at the minimum

This is just to document some of these things in order to make full covariances available in non-minuit backends @alexander-held

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[lukasheinrich]

Added a second notebook on errors (the calc in #764 ) and we see compatible errors with various versions of how to calculate them.

maybe a question to @alexander-held: is it important to your use-case to have MINUIT-like uncertainties or would uncertainties based on the standard covariance matrix (the blue dashed line) suffice? Separately from this we can implement MINOS-like uncert by doing a line search.

[lukasheinrich]

fwiw, this reproduces the errors in the 1Lbb lhood. (though the MINUIT specifics seem to not matter)

[alexander-held]

In the minuit_errors.ipynb notebook, the objective is 2 NLL, but errordef=0.5 - I think errordef=1 is needed if the objective is not the NLL.

[alexander-held]

I did not understand the last entry in the legend in the plots above, which seemed to have a completely different behavior. For me all the implementations look much more similar when using this notebook (and fixing index in the plotting, so the second parameter is shown):

I thought the discrepancy might be due to boundary effects, since the POI is limited to [0, 10] and hits both ends of this boundary (at 50 and 70 observed events).

I tried out

pdf = pyhf.simplemodels.hepdata_like([10.0], [500.0], [5.0])
scan = jnp.linspace(515, 545, 10)

which is further away from boundaries, but am surprised there is still a substantial difference:

Increasing all yields by a factor 10 (with constant background uncertainty) makes the discrepancy smaller, and with a factor 100 it slowly disappears.

Is the 1Lbb statistically limited? Maybe this is some systematics effect that causes differences? In any case I need to understand that other notebook better, since I did not expect such large differences.

[lukasheinrich]

the last entry (brown) is using minuit but supplying the AD gradient. This uses strategy one, so perhaps it's stoppinng eariler and therefore the underlying hesssian is different.

[alexander-held]

Thanks, the strategy was the difference between our setups. I was running with 0.5.4 which does not yet include the switch to strategy 0 if gradients are provided. I am surprised that the difference is this large. I thought the HESSE call would help, but maybe the convergence fails completely? Strange since this likelihood is relatively simple.