Bayesian Priors

This project contains Gulliver likelihood services which add a bias (zenith weight) to reconstructions. These biased reconstructions do not produce a better estimate of the track direction, but rather are useful for distinguishing up-going tracks for down-going tracks. This is refered to as a “Bayesian” reconstruction because multiplying a liklihood by a function which only depends on reconstruction parameters (as opposed to data) is reminiscent of a Baysian prior, not due to any correct application of Bayes’ Theroem.

Maintainer : Kevin Meagher

Bayesian Priors

Bayes law states:

\[P(H|E) = \frac{P(E|H)P(H)}{P(E)}\]

Where \(H\) is the track hypothesis and \(E\) is the hit pattern observed in the detector. \(P(E)\) is a normalization and can be ignored. \(P(E|H)\) is the probability of observing the hit pattern given a track hypothesis and \(P(H)\) is a probability of observing a track \(H\) independent of the observed hit pattern \(E\). The likelihood, \(\Lambda(E;H)\) from IceCube reconstructions can be substituted for \(P(E|H)\), thus multiplying a reconstruction likelihood by a probability which depends only on \(H\) is referred to as a bayesian reconstruction.

\(P(H)\) is referred to as a prior probability. In Bayesian inference, the prior probability distribution is probability that would express one’s beliefs about this quantity before some evidence is taken into account. In this context it is the probability of observing a track completely independent of of the hit pattern observed in the detector. Adding the prior to the adds a substantial bias to reconstruction, but greatly improves the ability to distinguish up-going tracks from down-going tracks, especially near the horizon.

This technique was first developed by the NEVOD neutrino detector collaboration who were able to extract an atmospheric neutrino from a background of 1010 atmospheric muons in a small (6×6times×7.5 m3) neutrino detector on the surface[1]. This technique was independently developed for the AMANDA experiment[2][3][4].

Pure Bayesian priors

The pure Bayesian priors (in IceCube/AMANDA reconstruction) try to model the zenith distribution of downgoing muons (bundles) induced by cosmic rays, and/or atmospheric neutrinos. We have currently three functions which attempt to model the zenith distribution of downgoing muons:

These function represent a Zenith distribution of down-going cosmic rays as observed by the AMANDA detector[5]. The atmospheric neutrino flux is modeled as a constant value all over sky. The atmospheric neutrino flux is not constant as a function of zenith but it is close enough that this approximation should be fine

In this purely Bayesian approach, an up-going track should only be found by the minimizer if its regular likelihood is so good that it is found by the minimizer even with the bias terms added (which favor down-going tracks), i.e. its likelihood should be about five or six orders of magnitude better than the best down going track.

However, it turns out that also a significant number of downgoing corsika tracks “survive” this reconstruction as an upgoing track.

Bayesian priors to restrict phase space

Bayesian priors can also be used to enforce e.g. a particular zenith range for the reconstruction, e.g. with the I3ConstantZenithWeightServiceFactory prior. This can also be achieved in other ways, for instance via the definition of the range of minimizer parameters, but this then also necessitates fixes in the seed generation. The phase space selection used to be done with a constant large positive penalty term to \(-\log(\Lambda)\), but it turns out that this still results in solutions in the “forbidden zone”, because of local minima. A different approach is to apply a penalty term which a very steep slope towards the allowed zone, so that whenever the seed track or a minimizer-generated track is in the forbidden zone, it will always “roll” back to the allowed zone of phase space (think of the minimizer solution as a marble rolling in a hilly log-likelihood landscape).

For all zenith weights implemented in this project it is possible to specify both a constant penalty value \(c\) and a “slope” value \(s\). In the forbidden range the penalty is then \(c+s*abs(CosZenDiff)\), where \(CosZenDiff\) denotes the difference in \(\cos(zenith)\) between the \(\cos(zenith)\) of the current event hypothesis and the closest allowed \(\cos(zenith)\) value.

Both

What turns out to work quite well for finding good upgoing muon tracks is to define a reconstruction in which we simply assume that the event was a down going muon, to apply both the zenith weight which models the CR spectrum (e.g. I3PowExpZenithWeightServiceFactory), as well as the phase space enforcing terms, i.e. a sloped penalty term for upgoing tracks. The likelihood difference of this reconstruction and an unbiased reconstruction of the same event, with an upgoing result, can be used as a discriminator between true upgoing events and downgoing events misreconstructed as upgoing.

Bayesian priors usage

This project only implements the priors. To use them in a Gulliver reconstruction, you instantiate them as a service in the tray with an tray.AddService(...) line in your python script, just like you do for your regular likelihood function (e.g. I3RecoLLHFactory). Then you combine the prior and the regular likelihood function using the I3EventLogLikelihoodCombinerFactory service from the gulliver project. See the bayesianfit.py example script under resources/scripts to get the idea.

Use as a Quality Cut

In IceCube, the Bayesian reconstruction is used distinguish, up-going tracks from down-going tracks, specifically to calculate a quality parameter for cutting background in up-going muon analyses, either as a direct cut or fed to a BDT. The so-called bayesratio is calculated as \(\log(\Lambda_{bayes})-\log(\Lambda_{normal})\). Typal cut values for bayesratio are around 25-30[6].

Avoiding Inappropriate Usage

The Bayes Reconstruction is a biased estimator of a muon tracks location, which means its expectation value is not equal to its true value. Thus, it is highly undesirable for use as the best guess for the location of a track. For upgoing tracks, the Bayesian reconstruction is often very far from the unbiased reconstruction, it is unlikely to be to be mistaken as a good reconstruction. However, for down-going only analyses the bias is small but still present.

Recall from statistics that estimators are usually judged on three criteria: consistancy, bias, and variance. It it a widly know property that in some cases a biased estimator will have a smaller variance than an unbiased estimator[7]. In the case of the Bayesian reconstruction the zenith distribution is biased and the variance of the zenith distribution in smaller than for a non-zenith weighted reconstruction. Since the parameter deltaangle, the average angular distance between the Monte Carlo truth direction and the reconstructed direction, is essencially the combination of the zenith variance and the azimuth variance, it too will be smaller than the deltaangle for the unbiased reconstruction. This situation has led to more than one IceCube collaborator trying to use Bayesian reconstructions in a down-going analysis on the sole justification that deltaangle was smaller. Or even worse, incorrectly claiming that the Bayesian reconstruction was more accurate because it incorperates the prior distribution, which contains more information. Please do not make either of these mistakes. You almost certainly do not want to use a biased reconstruction.

See Also

References