Please revise get_wstat_mu_bkg about gammapy HOT 10 CLOSED

Thanks @moralejo . I remember that this changed at some point to be consistent (see e.g. #1936).
Let's put this on the list for the next release.

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Hi,

the code is correct, but it returns the Poisson parameter of the background in the OFF region

Whereas the same formula in the documentation:
https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html

is used to obtain the Poisson parameter of the background in the ON region - that is what mu_bkg means in the documentation. So probably only the docs are wrong.

Here:

the return of the function (for the whole OFF region) is scaled by alpha, to get the value for the ON region, so it seems correct.
It would anyway be worth that an expert checks it. I also think it would be better to use, for the background Poisson parameters, variable names which make it clear straightaway whether they refer to the ON region or to the OFF region.

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Thanks @moralejo! Which modifications would you propose to https://docs.gammapy.org/1.1/user-guide/stats/index.html#notations, to make the description more clear?

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Thanks @moralejo! Which modifications would you propose to https://docs.gammapy.org/1.1/user-guide/stats/index.html#notations, to make the description more clear?

Hi,

I would say

"mu_bkg: predicted background counts in the ON region" ( this is what it means here, since below we have mu_off = mu_bkg / alpha)

Then, for consistency, the function get_wstat_mu_bkg should either be called get_wstat_mu_off , or be modified to return mu_bkg (and the code using the function should be changed as well).

In the current documentation https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html (not sure if everywhere else) mu_bkg has the same meaning as in the glossary - ok!
But then the formulae here:
https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html#profile-likelihood
need fixing (they were taken from a reference in which mu_bkg meant what in gammapy is called mu_noff). These would be the changes (crosscheck is welcome):

$\frac{\alpha,n_{\mathrm{on}}}{\mu_{\mathrm{sig}}+\alpha\mu_{\mathrm{bkg}}} + \frac{n_{\mathrm{off}}}{\mu_{\mathrm{bkg}}} - (\alpha+ 1) = 0 \longrightarrow \frac{\alpha \; n_{\mathrm{on}}}{\mu_{\mathrm{sig}}+\alpha\mu_{\mathrm{bkg}}} + \frac{\alpha \; n_{\mathrm{off}}}{\mu_{\mathrm{bkg}}} - (\alpha+ 1) = 0$

$\mu_{\mathrm{bkg}} = \frac{C + D}{2\alpha(\alpha + 1)} \longrightarrow \mu_{\mathrm{bkg}} = \frac{C + D}{2(\alpha + 1)}$
One can add $\mu_{bkg} = \alpha \mu_{off}$ for clarity, since in the code we will have: mu_bkg = alpha * get_wstat_mu_off(...)

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I think here:
https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html#special-cases

mu_bkg is used wrongly... it definitely should be mu_off in the likelihood formula.

Not sure if in the following lines it has consistently the same (wrong) meaning, I have to check it... Done, see below.

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There are also problems here:
https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html#goodness-of-fit

The Likelihood function is the same as before, with arguments:

So the correct arguments to get the absolute maximum likelihood are:
$L(n_{on}, n_{off}, \alpha, (n_{on} - \alpha \; n_{off}), \alpha \; n_{off})$

(and the same in the two formulae afterwards)

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In https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html#derivation-of-the-wstat-formula

Where it says "mu_src" it should be "mu_sig"

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https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html#final-result

https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html#special-cases

In the two formulae above, mu_bkg should become mu_bkg / alpha (or, equivalently, mu_off)

Should be: $L(0, n_{off}, \alpha, (0 - \alpha \; n_{off}), \alpha \; n_{off}) = ...$ (maximum absolute L in case n_on=0) The right side is correct.

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Last part...
Under https://docs.gammapy.org/1.1/user-guide/stats/wstat_derivation.html#special-cases:

Again, mu_bkg must be replaced above by mu_bkg / alpha

And

should become mu_bkg = [alpha / (alpha+1)] * n_off

(mu_bkg / alpha should replace mu_bkg above)

And the next one:

should rather be [alpha / (1+alpha)] n_on - mu_sig

After that, I think the rest is ok (the values of W)

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All that is just to make the doc page consistent with the meaning of mu_bkg in the glossary.

About the code: I saw no misuse of get_wstat_mu_bkg (the returned value is multiplied by alpha to get the actual mu_bkg). But I did not check anything else... it would be good if some expert can make sure that the computation of the goodness of fit is correct.

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