FACIES DE LOS DEPÓSITOS DE AVALANCHA DE

The presence of backgrounds means that the measured number events of N 6= Ns, but rather N = Ns+P_iNbi. The addition of a background process adds a new, distinct class of events

to the analysis besides the signal event class. Each background Ni

b is a random variable with

its own Poisson probability distribution, with expected value nb,i, with its own distributions for

the observables. The fact that the signal and background classes have different shapes for their invariant mass distributions provides a way for the number of events in each class to be extracted by the maximum likelihood method. For simplicity of notation let’s also call the expected number of signal events ns and define the expected number of total events to be n = ns+Pinb,i. The

likelihood changes in the following way

L(N, m4`|ns, nb,i) = Pois(N|n) events Y j ns nFs(m j 4`) + X i nb,i n Fb,i(m j 4`) ! (9.3)

WhereFsandFb,iare the probability distribution functions (PDFs) for them4`shape of both the

will not. If we continue to imagine that we had no systematic uncertainties and no acceptance and efficiency loss, thenns=σLint still, and is the only parameter we care to extract. Though the

likelihood depends on the background parameters, all of thenb,iare considered nuisance parameters,

which are parameters we are not interested in extracting but which add additional degrees of freedom to the analysis. The background parameters are not completely free parameters either, but are constrained to float between certain values, as will be explained later. To deal with the nuisance parameters when extracting the best fit value and confidence interval forns, we will use the profile

likelihood ratio fit [145].

The profile likelihood ratio is defined as

λ(ns) = L(N, m4`|ns, d d nb,i) L(N, m4`|cns,ndb,i) (9.4)

The denominator is the likelihood function at the global maximum where all the parameters have been set to their maximum likelihood estimate (MLE). In the numerator, d

d

nb,i are the MLEs

of the nuisance parameters for a given value ofns, i.e. for each different value ofns, the likelihood

is re-maximized at that point to find the MLE of the nuisance parameters. λ(ns) is now a function

of one variable, equal to 1 at the global maximum and 1> λ_≥0 at any other other value ofns.

In order to make good use of some asymptotic properties of the profile likelihood ratio, we will transform the variable to the -2 log likelihood,₋2 lnλ(ns).

Minimizing −2 lnλ(ns) will find the best estimate for σb = c

ns

Lint, while getting the confidence interval requires calculating an approximation of the PDF of−2 lnλ(ns). −2 lnλ(ns) is a random

variable, since it is a function of the random variables N and m4`. To approximate the PDF, we

use Wilk’s Theorem [136] which states that in the limit of largeN, the variable₋2 lnλ(ns) will be

distributed like aχ2_{variable with one degree of freedom. For aχ}2_{distributions,χ}2_{= 1 corresponds}

to a confidence level of 68.3% whileχ2_{= 4 corresponds to a confidence level of 95.5%.}

In order to calculate the 68% confidence interval then, it is necessary to find the values of n∗ s

such that ₋2 lnλ(n∗

s) + 2 lnλ(ncs) = 1. When doing profile likelihood ratio minimization with

ROOTMINUIT[144], callingMINOSwill find then∗

s(both the lower and upper bound) which return

−2 lnλmin+ 1. MINOSfollows the−2 lnλ(ns) function out from the global minimum to find where

it crosses +1 value, instead of using the curvature at the global minimum and assuming a parabolic shape, like withHESSE.

9.2.1

Adding Systematic Uncertainties

When calculating the estimate of the cross section, we have to divide out the integrated luminosity from_cns. The integrated luminosity is not a constant, but a measured quantity with its own uncer-

tainty. To take into account the uncertainty of the luminosity, we add a Gaussian constraint to the likelihood in the form of

L(N, m4`|ns, nb,i, θL) = Pois(N|n)×P(mj4`|s, nb,i)×Gaus(ΘL|θL, αL) (9.5) P(mj<sub>4`|</sub>s, nb,i) = events Y j s nFs(m j 4`) + X i nb,i n Fb,i(m j 4`) ! (9.6)

Θ_L is a Gaussian random variable with meanθ_L and standard deviationα_L. Θ_L represents the percent deviation from the measured value of the luminosity, and so in the likelihood is Θ_L = 0. During the fit, the parameterθL, the mean deviation, is allowed to float in order to find the global

best fit, and may not necessarily be 0. It also acts as a constraint when doing a maximum likelihood fit, as values too far away from 0 become excessively ’unlikely’. αL is always fixed and is gotten

from experimental measurements. The number of expected signal events in the likelihood are now constructed as

s=ns(1 +θL·αL) =σLint(1 +θL·αL) (9.7) nsis the nominal number of expected signal events, andsis the expected number of signal events

taking into account variation due to the uncertainty on the luminosity. The notation change in the likelihood fromns→sreflects this difference. The value ofLint is fixed while its variation is taken

care of by the parameterθL. When doing the profile likelihood ratio fit,θLandnb,iare the nuisance

parameters which are constrained by experimental data, whileσ is the parameter of interest to be estimated. Adding more nuisance parameters has the tendency to widen the confidence interval gotten from the profile fit.