654
L. Lyons
15.2.3 Profile Likelihood
In many situations the likelihood is a function not only of the parameter of interest φ
but also other parameters. These may be other physics parameters (for example, in
neutrino oscillation experiments where the mixing angles and differences in masssquared of the various neutrinos are relevant), but can also be nuisance parameters
ν associated with systematic effects (e.g. jet energy scales, particle identification
efficiencies, etc.). To make statements about φ, the likelihood L(φ.ν) is often
‘profiled’ over the nuisance parameters, i.e. at each value of φ, the likelihood is
remaximised with respect to ν. Thus
L prof (φ) = L(φ, ν max (φ))
(15.4)
Then L prof (φ) is used much as the ordinary likelihood when there are no nuisance
parameters.
A profile likelihood is in general wider than the likelihood for a fixed value of
the nuisance parameter ν; this results in the uncertainty in the parameter of interest
φ being larger when allowance is made for the systematic uncertainties.
In the standard profile likelihood, ν is a continuous variable. An extension of this
has been used by Dauncey et al. [38], to allow for uncertainties in the choice of
functional form of the background parametrisation in searches for new particles as
peaks above background in a mass spectrum. Here the systematic is discrete, rather
than continuous.
An alternative way of eliminating nuisance parameters (known as marginalisation) is to use L(φ, ν) as part of a Bayesian procedure, and than to integrate the
Bayesian posterior over ν. i.e.
P marg (φ) =
P post (φ, ν) dν
(15.5)
Of course, both profiling and marginalisation result in the loss of information.
Reference [37] provides a very trivial example of this for profile likelihoods.
15.2.4 Punzi Effect
Sometimes we have two or more nearby peaks, and we try to fit our data in order
to determine the fractions of each peak. Punzi [39] has pointed out that it is very
easy to write down a plausible but incorrect likelihood function that gives a biassed
result. This occurs in situations where the events have experimental resolutions σ
in the observable x that vary event-by-event; and the distributions of σ are different
for the two peaks.
L. Lyons
15.2.3 Profile Likelihood
In many situations the likelihood is a function not only of the parameter of interest φ
but also other parameters. These may be other physics parameters (for example, in
neutrino oscillation experiments where the mixing angles and differences in masssquared of the various neutrinos are relevant), but can also be nuisance parameters
ν associated with systematic effects (e.g. jet energy scales, particle identification
efficiencies, etc.). To make statements about φ, the likelihood L(φ.ν) is often
‘profiled’ over the nuisance parameters, i.e. at each value of φ, the likelihood is
remaximised with respect to ν. Thus
L prof (φ) = L(φ, ν max (φ))
(15.4)
Then L prof (φ) is used much as the ordinary likelihood when there are no nuisance
parameters.
A profile likelihood is in general wider than the likelihood for a fixed value of
the nuisance parameter ν; this results in the uncertainty in the parameter of interest
φ being larger when allowance is made for the systematic uncertainties.
In the standard profile likelihood, ν is a continuous variable. An extension of this
has been used by Dauncey et al. [38], to allow for uncertainties in the choice of
functional form of the background parametrisation in searches for new particles as
peaks above background in a mass spectrum. Here the systematic is discrete, rather
than continuous.
An alternative way of eliminating nuisance parameters (known as marginalisation) is to use L(φ, ν) as part of a Bayesian procedure, and than to integrate the
Bayesian posterior over ν. i.e.
P marg (φ) =
P post (φ, ν) dν
(15.5)
Of course, both profiling and marginalisation result in the loss of information.
Reference [37] provides a very trivial example of this for profile likelihoods.
15.2.4 Punzi Effect
Sometimes we have two or more nearby peaks, and we try to fit our data in order
to determine the fractions of each peak. Punzi [39] has pointed out that it is very
easy to write down a plausible but incorrect likelihood function that gives a biassed
result. This occurs in situations where the events have experimental resolutions σ
in the observable x that vary event-by-event; and the distributions of σ are different
for the two peaks.