[MuonGun] Implement unbiased generation

[Jakob van Santen]

Coincident background injection requires truly unbiased generation. Currently this is not directly supported; the closest you can get is drawing energies from a spectrum that is close to the natural spectrum. Add a rejection sampling (or similar) scheme to draw bundle energies directly from the parameterized flux.

[Jakob van Santen]

In 148890/IceCube:

Work in progress from a few months ago to implement natural-rate sampling (see
#1399). Things done so far:

• Correct implementation of MUPAGE parameterizations. It turns out that the energy units were wrong. Durr.
• Tentative implementation of joint sampling from the energy/radius distribution. For BMSS (MUPAGE) this is direct; for the spline distributions it uses an indepencence Metropolis-Hastings sampler with BMSS as the proposal distribution.

The latter is unsatisifying, as the sampler is not guaranteed to converge to
the stationary distribution, nor is the correlation between the samples known.
The obvious approach of sampling first from the unconditional radial
distribution (i.e. integrated over all energies) does not work in general,
because the parameterized radial distibution does not match the integral of the
joint distribution everywhere (they were fit separately). Straight rejection
sampling is also not great, since the location of the maximum is not generally
known, nor can one guarantee that the surface is either strictly log-concave or
log-convex. More thought (or perhaps acceptance of general slowness) is needed.

[Jakob van Santen]

In 148992/IceCube:

Implement sampling from joint radius/energy distribution via affine-invariant
ensemble MCMC (cribbed from emcee). Next step is to work up an injector that
knows how to use natural-rate bundles. See #1399.

[Jakob van Santen]

Natural rate sampling is now implemented in branches/unbiased-generation. Now someone needs to test whether it produces sane-looking data.

[Juan Carlos Díaz Vélez]

This is awesome! I will test.

[Alex Olivas]

Testing this might be a natural candidate for the Strike Team post-Stockholm.

[Jakob van Santen]

This still needs to be compared to CORSIKA-based coincident background generation.

[Alex Olivas]

Apparently this in on the trunk already and Juan Carlos is going to generate a set of unbiased MuonGun? samples. Once that's done we'll close this as fixed and open another tracking ticket for the comparisons, which will likely take special care and be a longer term project.

[Jakob van Santen]

Comparisons have been attempted in the past, but they were hampered by poorly chosen generator settings. Since you can't easily generate CORSIKA that only produces a specific range of muon energies at depth, the MuonGun? simulation needs to cover the full muon energy range of the parameterization. Likewise, since MuonGun? integrates over a fixed primary composition, the CORSIKA set you compare to needs to cover the full range of primary energies that contribute to the muon flux at depth.

[Alex Olivas]

It's time to start thinking about the next milestone. Going to try something new, since waiting until the code sprints to attack tickets hasn't provided the results we were hoping for.

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[Alex Olivas]

To the owner of this ticket. Signal you're aware if this ticket, by 'accept'ing it.

[Alex Olivas]

Moved to smartsheets.