5.10 Statistical Evaluation of Modeling Errors and Quality of Predictions in. . .
229
a global scaling of the initial adopted errors:
δ ¯
O i → δ ¯
O i
χ 2 (p 0 )/N dof .
(5.76)
With this choice of the normalization condition, one can apply the standard rules
of linear regression in statistical analysis and assess quantities such as the covariance
matrix and statistical uncertainties. The Jacobian J associated to χ 2 (p) and its
Hessian matrix ˆ ¯
H are straightforward to calculate from Eq. (5.74):
ˆ ¯
J =
1
δ ¯
O i
∂ ¯
O i
∂p j
(p)
p=p 0
ij
(5.77)
ˆ ¯
H =
∂ 2 χ 2
∂p i ∂p j
(p)
p=p 0
ij
= ˆ ¯
J
T ˆ ¯
J ,
(5.78)
where the two first equations are valid in the general case whereas the last equation
pertains only to the linear approximation (see Exercise VII).
Exercise VII
Derive Eq. (5.78) by differentiating Eq. (5.74).
The probability distribution verified by p is somewhat arbitrary and has to be
postulated based on physical grounds. One can demand that it has a Gaussian
dependence with respect to χ 2 (p) [50]:
¯
P (p) ∝ exp(−χ
2 (p)) ,
(5.79)
where its overall factor is determined by normalization.
Equation (5.79) can be understood intuitively: if χ 2 (p) is very sharp around the
minimum for some parameters, the latter are unlikely to be much different in the
optimal set of parameters because modifying them would provide with results of
poor quality for the already fitted observables. Conversely, if χ 2 (p) has a flat bottom
therein, these parameters are sloppy and can significantly vary if other observables
are added to the fitting procedure, for example. The covariance matrix ¯
C(p 0 ) can
then be expressed in terms of the Hessian matrix ˆ
H and hence Jacobian J of
Eqs. (5.77) and (5.78):
¯
C(p 0 ) = ˆ ¯
H
−1 = ( ˆ ¯
J
T ˆ ¯
J )
−1 .
(5.80)
229
a global scaling of the initial adopted errors:
δ ¯
O i → δ ¯
O i
χ 2 (p 0 )/N dof .
(5.76)
With this choice of the normalization condition, one can apply the standard rules
of linear regression in statistical analysis and assess quantities such as the covariance
matrix and statistical uncertainties. The Jacobian J associated to χ 2 (p) and its
Hessian matrix ˆ ¯
H are straightforward to calculate from Eq. (5.74):
ˆ ¯
J =
1
δ ¯
O i
∂ ¯
O i
∂p j
(p)
p=p 0
ij
(5.77)
ˆ ¯
H =
∂ 2 χ 2
∂p i ∂p j
(p)
p=p 0
ij
= ˆ ¯
J
T ˆ ¯
J ,
(5.78)
where the two first equations are valid in the general case whereas the last equation
pertains only to the linear approximation (see Exercise VII).
Exercise VII
Derive Eq. (5.78) by differentiating Eq. (5.74).
The probability distribution verified by p is somewhat arbitrary and has to be
postulated based on physical grounds. One can demand that it has a Gaussian
dependence with respect to χ 2 (p) [50]:
¯
P (p) ∝ exp(−χ
2 (p)) ,
(5.79)
where its overall factor is determined by normalization.
Equation (5.79) can be understood intuitively: if χ 2 (p) is very sharp around the
minimum for some parameters, the latter are unlikely to be much different in the
optimal set of parameters because modifying them would provide with results of
poor quality for the already fitted observables. Conversely, if χ 2 (p) has a flat bottom
therein, these parameters are sloppy and can significantly vary if other observables
are added to the fitting procedure, for example. The covariance matrix ¯
C(p 0 ) can
then be expressed in terms of the Hessian matrix ˆ
H and hence Jacobian J of
Eqs. (5.77) and (5.78):
¯
C(p 0 ) = ˆ ¯
H
−1 = ( ˆ ¯
J
T ˆ ¯
J )
−1 .
(5.80)