228
5 Formulation and Implementation of the Gamow Shell Model
model averaging. In the context of Gamow shell model, model averaging consists
of evaluating the same observable quantity with different sets of parameters in the
interactions used so as to minimize the error made on the observables arising from
all made calculations.
The Bayesian approach has been implemented in density functional theory,
where it is used to assess the errors made on the parameters of used Skyrme
interactions along with model averaging by comparing the results obtained in
density functional theory with those from a finite-range liquid drop model [46].
The Bayesian analysis has also recently been applied to reaction theory [47, 48],
where the theoretical uncertainties in calculated cross sections can be calculated
from the used fitted optical potentials. While the Bayesian analysis has not been yet
fully implemented in the Gamow shell model, introductory calculations have shown
that it can provide with more efficient estimators of theoretical statistical errors of
observables in the Gamow shell model.
5.10.1 Penalty Function and Linear Regression Approximation
The optimization of the interaction and the assessment of statistical uncertainties
is performed according to Refs. [49, 50]. Given a model in which N p parameters
p = {p 1 , ., p N p } are adjusted to describe N d observables ¯
O i (i = 1, ., N d ), the
optimization procedure is based on the minimization of the penalty function:
χ
2 (p) =
1
2
N d
i=1
¯
O i (p) − ¯
O
exp
i
δ ¯
O i
2
,
(5.74)
where ¯
O i (p) are the calculated observables and ¯
O
exp
i are the experimental data (fitobservables) used to constrain the model. The adopted errors δ ¯
O i include different
contributions stemming from experimental uncertainties, numerical inaccuracies,
and theoretical errors due to model deficiency.
The choice for the theoretical error obviously involves a certain level of
arbitrariness even when driven by physical considerations. Part of this arbitrariness
can be removed by tuning the adopted errors so that they are consistent with the
distribution of the residuals similarly to the case of a purely statistical distribution
[50]. In particular, one requires the total penalty function to be normalized to the
number of degrees of freedom N dof = N d − N p at the minimum p 0 [51]:
χ 2 (p 0 )
N dof
↔ 1 .
(5.75)
In the case of a single type of data and on the assumption that experimental and
numerical errors are negligible, the condition (5.75) can simply be achieved through
5 Formulation and Implementation of the Gamow Shell Model
model averaging. In the context of Gamow shell model, model averaging consists
of evaluating the same observable quantity with different sets of parameters in the
interactions used so as to minimize the error made on the observables arising from
all made calculations.
The Bayesian approach has been implemented in density functional theory,
where it is used to assess the errors made on the parameters of used Skyrme
interactions along with model averaging by comparing the results obtained in
density functional theory with those from a finite-range liquid drop model [46].
The Bayesian analysis has also recently been applied to reaction theory [47, 48],
where the theoretical uncertainties in calculated cross sections can be calculated
from the used fitted optical potentials. While the Bayesian analysis has not been yet
fully implemented in the Gamow shell model, introductory calculations have shown
that it can provide with more efficient estimators of theoretical statistical errors of
observables in the Gamow shell model.
5.10.1 Penalty Function and Linear Regression Approximation
The optimization of the interaction and the assessment of statistical uncertainties
is performed according to Refs. [49, 50]. Given a model in which N p parameters
p = {p 1 , ., p N p } are adjusted to describe N d observables ¯
O i (i = 1, ., N d ), the
optimization procedure is based on the minimization of the penalty function:
χ
2 (p) =
1
2
N d
i=1
¯
O i (p) − ¯
O
exp
i
δ ¯
O i
2
,
(5.74)
where ¯
O i (p) are the calculated observables and ¯
O
exp
i are the experimental data (fitobservables) used to constrain the model. The adopted errors δ ¯
O i include different
contributions stemming from experimental uncertainties, numerical inaccuracies,
and theoretical errors due to model deficiency.
The choice for the theoretical error obviously involves a certain level of
arbitrariness even when driven by physical considerations. Part of this arbitrariness
can be removed by tuning the adopted errors so that they are consistent with the
distribution of the residuals similarly to the case of a purely statistical distribution
[50]. In particular, one requires the total penalty function to be normalized to the
number of degrees of freedom N dof = N d − N p at the minimum p 0 [51]:
χ 2 (p 0 )
N dof
↔ 1 .
(5.75)
In the case of a single type of data and on the assumption that experimental and
numerical errors are negligible, the condition (5.75) can simply be achieved through