15 Statistical Issues in Particle Physics
655
For a set of observations x i , it is tempting but wrong to write the unbinned
likelihood as
L(f ) wrong = {f ∗ G(x i , 0.0, σ i ) + (1 − f ) ∗ G(x i , 1.0, σ i )}
(15.6)
where f is the fraction of the first peak (labelled A below) which is parametrised
as G(x i , 0.0, σ i ), a Gaussian in x i , centred on zero, and with width σ i , and i is the
label for the ith event; and similarly for the second peak (labelled B), except that it
is centred at unity.
Application of the rules of conditional probability shows that the correct
likelihood is
L(f ) right = {f ∗ G(x i , 0.0, σ i ) ∗ p(σ i |A) + (1 − f ) ∗ G(x i , 1.0, σ i ) ∗ p(σ i |B)}
(15.7)
where p(σ i |A) and p(σ i |B) are the probability densities for the resolution being
σ i for the A and B peaks respectively. We then see that L(f ) wrong and L(f ) right
give identical values for f , provided that p(σ i |A) = p(σ i |B). If however, the
distributions of the resolution differ, L(f ) wrong will in general give a biassed
estimate.
Punzi investigated the extent of this bias in a simple Monte Carlo simulation, and
it turns out to be surprisingly large. For example, with f = 1/3, and p(σ A ) and
p(σ B ) being δ−functions at 1.0 and at 2.0 respectively (i.e. σ = 1 for all A events,
and σ = 2 for all B events), the fitted value of f from L(f ) wrong turned out to be
0.65. Given that f is confined to the range from zero to unity, this is an enormous
bias.
The way the bias arises can be understood as follows: The fraction f of the
events that are really A have relatively good resolution, and so the fit to them alone
would assign essentially all of them as belonging to A i.e. these events alone would
give f ≈ 1 with a small uncertainty. In contrast the 1 − f of the events that are
B have poor resolution, so for them the fit does not mind too much what is the
value of f . But the fit uses all the events together, and so assigns a single f to the
complete sample; this will be a weighted average of the f values for the A and for
the B events. Because the A events result in a more accurate determination of f
than do the B events, the fitted f will be biassed upwards (i.e. it will over-estimate
the fraction of events corresponding to the peak with the better resolution).
The Punzi effect can also appear in other situations, such as particle identification. Different particle types (e.g. pions and kaons) would appear as different peaks
in the relevant particle-identification variable e.g. time of flight, rate of energy loss
dE/dx, angle of Cherenkov radiation, etc. The separation of these peaks for the
different particle types depends on the momentum of the particles (see Fig. 15.2).
The incorrect L is now
L wrong (f K ) = {(1−f K )∗G(x i , x π (p i ), σ i )+f K ∗G(x i , x K (p i ), σ i )} (15.8)
655
For a set of observations x i , it is tempting but wrong to write the unbinned
likelihood as
L(f ) wrong = {f ∗ G(x i , 0.0, σ i ) + (1 − f ) ∗ G(x i , 1.0, σ i )}
(15.6)
where f is the fraction of the first peak (labelled A below) which is parametrised
as G(x i , 0.0, σ i ), a Gaussian in x i , centred on zero, and with width σ i , and i is the
label for the ith event; and similarly for the second peak (labelled B), except that it
is centred at unity.
Application of the rules of conditional probability shows that the correct
likelihood is
L(f ) right = {f ∗ G(x i , 0.0, σ i ) ∗ p(σ i |A) + (1 − f ) ∗ G(x i , 1.0, σ i ) ∗ p(σ i |B)}
(15.7)
where p(σ i |A) and p(σ i |B) are the probability densities for the resolution being
σ i for the A and B peaks respectively. We then see that L(f ) wrong and L(f ) right
give identical values for f , provided that p(σ i |A) = p(σ i |B). If however, the
distributions of the resolution differ, L(f ) wrong will in general give a biassed
estimate.
Punzi investigated the extent of this bias in a simple Monte Carlo simulation, and
it turns out to be surprisingly large. For example, with f = 1/3, and p(σ A ) and
p(σ B ) being δ−functions at 1.0 and at 2.0 respectively (i.e. σ = 1 for all A events,
and σ = 2 for all B events), the fitted value of f from L(f ) wrong turned out to be
0.65. Given that f is confined to the range from zero to unity, this is an enormous
bias.
The way the bias arises can be understood as follows: The fraction f of the
events that are really A have relatively good resolution, and so the fit to them alone
would assign essentially all of them as belonging to A i.e. these events alone would
give f ≈ 1 with a small uncertainty. In contrast the 1 − f of the events that are
B have poor resolution, so for them the fit does not mind too much what is the
value of f . But the fit uses all the events together, and so assigns a single f to the
complete sample; this will be a weighted average of the f values for the A and for
the B events. Because the A events result in a more accurate determination of f
than do the B events, the fitted f will be biassed upwards (i.e. it will over-estimate
the fraction of events corresponding to the peak with the better resolution).
The Punzi effect can also appear in other situations, such as particle identification. Different particle types (e.g. pions and kaons) would appear as different peaks
in the relevant particle-identification variable e.g. time of flight, rate of energy loss
dE/dx, angle of Cherenkov radiation, etc. The separation of these peaks for the
different particle types depends on the momentum of the particles (see Fig. 15.2).
The incorrect L is now
L wrong (f K ) = {(1−f K )∗G(x i , x π (p i ), σ i )+f K ∗G(x i , x K (p i ), σ i )} (15.8)