Unified Monte Carlo: Nuclear System Analysis
UMC
: Unfinished Business
Donald L. Smith
(ANL, retired)
Roberto Capote (IAEA)
Denise Neudecker (LANL)
CSEWG Meeting
2 – 4 November 2015
?
UMC
Unified Monte Carlo
The Concept
•
Assume that
knowledge
about a set of nuclear
observable
parameters
employed in nuclear system analyses can be
represented by a multi-variate master
probability function
.
•
This function should be
constructed
by incorporating the best
available information from
theory
and
experiments
.
•
The master function is then sampled using
Monte Carlo
methods
to generate a
Markov Chain
of
random
observable parameter
vectors
that ultimately can be employed for a variety of
practical
applications
such as generating
evaluations
and analyzing the
behavior
of derived nuclear
system parameters
.
Master Probability Density Function
Bayes Theorem:
p
(
x
|
T
,
E
) =
p
0
(
x
|
T
) L(
x
|
T
,
E
)
•
“
T
” signifies
prior
information based on
theory
(modeling).
•
“
E
” denotes
independent
information from
experiments
that
serves to
improve
(or augment) prior theoretical knowledge
T
.
•
“
x
” represents
random vectors
corresponding to possible values
of the nuclear
observables
(e.g., cross sections).
•
The
prior
probability function
p
0
is based on
theory
, while
likelihood
L
is a probability function that
quantifies
the
consistency
of data from
theory
and
experiments
used to
construct the master (posterior) probability function
p
(
x
|
T
,
E
).
Prior Probability Function
p
0
(
x
|
T
)
•
T
is a complicated
algorithm
that
maps
theoretical
model
parameters
q
to
calculated
observables
x
, i.e.,
x
=
T
(
q
).
•
By applying Monte Carlo techniques, a Markov Chain of
vectors
q
k
can be generated by
random sampling
of
parameters in space
S
(
q
)
governed by a
probability
function
r
0
(q)
. Usually,
mean values
q
0
and
covariance matrix
V
q
are specified. Then,
Maximum Entropy
suggests
r
0
should be a
normal probability
function.
•
A Markov chain of
values
x
k
in
observables space
S
(
x
) is generated
by Monte Carlo sampling according to
x
k
=
T
(
q
k
).
•
The collection {x
k
}
reflects
the
prior probability
function
p
0
, but
rarely
(if ever) can
p
0
be
expressed
explicitly
as an
analytical
function
that can be sampled in a conventional way!
S
{x
}
S
{q
}
q
k
p
0
(
x
|
T
) > 0
L(
x
|
T
,
E
) > 0
p
0
(
x
|
T
)
≈
0
L(
x
|
T
,
E
) ≈ 0
·
·
x
k
r
0
(
q
) > 0
x
k
=
T
(
q
k
)
p
0
(
x
|
T
) and L(
x
|
T
,
E
) > 0
so
p
(
x
|
T
,
E
) > 0
r
0
(
q
)
≈
0
Topology Issues
The
schematic diagram
shows
mapping
from space
S
{q
} to space
S
{x
} by the
theoretical
(model)
algorithm
T
. The
shaded
areas
denote regions of
non-negligible
probability
for
r
0
(green) and
p
0
(blue).
The region enclosed by a
red
dashed circle
indicates that portion of space
S
{x
} where the
likelihood
function
L(
x
|
T
,
E
) is
non-negligible
.
In the region labeled
Overlap
, where “blue”
and “red” dashed circles intersect, the
master
(posterior)
function
is also
non-negligible
.
UMC-G
: Analytical Approximation to
p
0
(
x
|
T
)
D.L. Smith,
Proceedings of AccApp’07
, Pocatello, ID, July 29 – August 2, 2007, Amer. Nucl. Soc. , p. 736.
•
The collection of K calculated observable parameter
vectors
{x
k
} generated by
Monte Carlo
(see preceding two slides), according to the mapping
x
k
=
T
(
q
k
), is
used to calculate
mean values
x
0
and
covariance matrix
V
x
via the formulas:
x
0i
≈
(∑
k=1,K
x
ik
) / K and (
V
x
)
ij
≈ [(∑
k=1,K
x
ik
x
jk
) / K] - x
0i
x
0j
(K is
very
large).
•
The “true”
prior probability
function
p
0
(
x
|
T
) typically is
approximated
by a
multi-variate
normal probability
function given by:
p
0
(
x
|
T
) ≈ C exp {-(½)[(x
–
x
0
)
T
V
x
-1
(
x
–
x
0
)]} (C is a normalization constant).
Advantage
: A lengthy Markov Chain of
sample values
is thus
replaced
by an
analytical approximation
having the
same mean values
and
covariance matrix
.
This yields a master (posterior) function
p
(
x
|
T
,
E
) that can be
sampled
readily
by
conventional Monte Carlo
methods, e.g., “Brute Force” or “Metropolis-Hastings”.
Disadvantage
: This approximation
discards
all information pertaining to
higher-
order
distribution
moments
inherent in the Monte Carlo generated Markov chain
{x
k
}. This
rejection
of information can lead to significant
biases
in cases where
non-linear
effects and distribution
skewness
and
kurtosis
are present.
UMC-B
: Information in
p
0
(
x
|
T
) is Preserved
R. Capote et al.,
Proceedings of ISRD-14
, Breton Woods, NH, May 22 – 27, 2011, ASTM STP-1550, p. 179.
•
The
collection
of K calculated observable parameter
vectors
{x
k
} generated by
Monte Carlo
(shown in two earlier slides), according to the mapping
x
k
=
T
(
q
k
), is
preserved
. Thus,
no information
on higher-order moments of
p
0
is
discarded
.
•
For each
x
k
, a scalar
weighting factor
ω
k
is generated according to the
expression:
ω
k
= L(
x
k
|
T
,
E
). Thus, the
worth
that is assigned to each MC sampled
parameter vector
x
k
is based on its
consistency
with available
experimental
data
, as reflected in the
likelihood
function.
•
For
very
large K, it is assumed that
mean values
and
covariance matrix
for the
master (posterior) probability function
p
(
x
|
T
,
E
) are estimated from:
x
0i
≈ (∑
k=1,K
ω
k
x
ik
) / (∑
k=1,K
ω
k
) and (
V
x
)
ij
≈ [(∑
k=1,K
ω
k
x
ik
x
jk
) / (∑
k=1,K
ω
k
)] - x
0i
x
0j
•
The
Markov Chain
for
UMC-B
thus consists of the set of pairs
{xk
,
ω
k
}
. These
values can be used for nuclear
systems applications
as well as
evaluations
.
Advantage
: All
information
in function
p
0
(x|
T
)
is clearly
preserved
, including that
related to
non-linearity
as well as the distribution
skewness
and
kurtosis
.
A Closer Look at UMC-G and UMC-B
L(
x
|
T
,
E
) > 0
p
0
(
x
|
T
) > 0
p
0
(
x
|
T
) and L(
x
|
T
,
E
) > 0
so
p
(
x
|
T
,
E
) > 0
•
The areas enclosed by “blue” and “red” dashed
circles, respectively, indicate
regions
of
non-
negligible probability
for the model-generated
prior probability
and the
likelihood function
that
quantifies
the
consistency
of theory and
experiment.
•
The
small
region
Overlap
of these two circles is
indicative of
data inconsistency
. Such an
outcome could have potentially
negative
implications
for an application of the
UMC-B
method (e.g.,
limited
or
biased sampling
of the
sparsely sampled region
Overlap
).
Statistical
inadequacy
is
not a problem
in applying the
UMC-G
approach, but it can
suffer
from
significant
bias effects
due to the explicit
rejection
of
higher-order moments
of
p
0
(
x
|
T
).
Data inconsistency
between theory and
experiment will inevitably lead to evaluations
and system analysis
results
that are very
questionable
and thus inherently
unreliable
.
Unfinished Business
•
Further investigation of the mentioned
sampling issues for the region
Overlap
in
the UMC-B approach is warranted.
•
Detailed inter-comparisons of GLS, UMC-
G, and UMC-B predictions for extreme
cases and inconsistent data are needed.
Yikes
! The region
Overlap
is very
small
,
probably due to poor
agreement between
theory and
experiments.
o
o
o
o
UMC-G Plus
: A New Option?
•
The
original UMC-G
formulation
discards
potentially valuable
information
about
higher moments
of the
prior probability
p
0
(
x
|
T
) that is
reflected
in the
Markov Chain of
vectors
{x
k
} generated by
Monte Carlo
sampling.
•
It is
unlikely
that the
moments
of
p
0
of
higher order
than mean values,
covariances, skewness, and kurtosis will
affect applications
significantly.
•
The
moments
of
p
0
(mean values, covariances, skewness, and kurtosis) can be
estimated
using the collection of
sample vectors
{x
k
}.
Suggestion
: Perhaps
analytical functions
might be found whose
parameters
can be
adjusted
to
approximate
the distribution
moments
deduced from {x
k
}. It could then
be employed to serve as a
surrogate
for
the
master
(posterior)
probability
function
p
(
x
|
T
,
E
), and it would then be
sampled
using
conventional Monte Carlo
methods.
Unfinished Business
•
Investigate the structure of realistic MC-
generated distributions
p
0
(
x
|
T
) with the
intent of quantifying typical mean values,
covariances, skewness, and kurtosis.
•
Identify families of analytical mathematical
functions that might serve as surrogates for
representing the MC-generated distributions
p
0
(
x
|
T
) with greater fidelity than using a
simple normal distribution (as in UMC-G).
Thoughts on Likelihood Functions L(
x
|
T
,
E
)
•
Available
experimental data
are usually comprised of
mean values
and (far less often)
covariances
. Therefore,
comparisons
between theoretically
calculated observables
and
experimental
observables
should
involve
at most
mean values
and
covariances
.
•
Consequently, the
likelihood
function L(
x
|
T
,
E
), in accordance with
Maximum Entropy
,
should be an appropriately constructed
normal probability
function. In particular, it
should have the form:
L(
x
|
T
,
E
) = C exp {-(½)[(y
–
y
E
)
T
V
E
-1
(
y
–
y
E
)]} (C is a normalization constant)
Note
:
y
E
is an
experimental data vector
with
covariance matrix
V
E
. Furthermore,
y
=
f
(
x
),
since what is measured (
y
)
may not correspond
directly to the
observable parameters
(
x
)
that are being considered. The function collection “
f
” establishes how
x
and
y
are related.
•
It may be very
difficult
to
construct
a rigorous
likelihood function
L(
x
|
T
,
E
) in any given
situation due to one or more of the following limitations: i)
incomplete data
, ii)
discrepant
(wrong)
data
, iii)
weak sensitivity
relationships between the data (
y
) and
parameters of interest (
x
), and iv) excessive
computational overhead
.
Alternatives
: Because of these
limitations
, some
investigators (notably
A. Koning
and
D. Rochman
)
for
pragmatic reasons
have investigated using
simpler
alternative likelihood
functions
L(
x
|
T
,
E
).
Unfinished Business
•
The impact of experimental data quality
and availability on applications of UMC
needs to be investigated thoroughly.
•
Improve experimental covariance data.
The UMC Approach at a Crossroads?
There are unresolved
technical
issues
and unanswered
questions
.
The
way forward
to further develop UMC must be
clarified
.
•
Would
UMC-G
and
UMC-B
be truly
comparable
if
p
0
(x|
T
)
could be
expressed exactly as an
analytical function
?
•
Can more
sophisticated
analytical function
approximations
to a
MC prior than the normal distribution be found (e.g.,
UMC-G Plus
)?
•
Are the available
experimental data
sufficiently
accurate
and
comprehensive
to be useful in practice for
applying UMC
?
•
Can
better
theoretical
models
be developed to
reduce
the
discrepancies
between
theory
and quality
experimental data
?
•
If not, can
model-defects
formalisms
be developed as practical
measures to
cope
with
model vs. experimental data
discrepancies
?
•
How much extra
“value”
does
UMC
contribute
,
compared
with
GLSQ
, to justify the additional
effort
and
computational burden
?
The End
In the field of nuclear system analyses, the concept of Unified Monte Carlo involves constructing a multi-variate master probability function using theory and experiment data. This function is then sampled to generate random observable parameter vectors for practical applications such as evaluations and system parameter analysis. Bayes Theorem plays a significant role in relating prior theoretical knowledge with independent experimental data to form the posterior probability function. Topology issues in mapping theoretical algorithms to observable spaces are also addressed, ensuring non-negligible probability regions are considered. The process involves complex algorithms and Monte Carlo techniques to generate Markov chains for both theoretical model parameters and calculated observables.
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Presentation Transcript
UMC: Unfinished Business Donald L. Smith (ANL, retired) Roberto Capote (IAEA) Denise Neudecker (LANL) CSEWG Meeting 2 4 November 2015 ?
UMC Unified Monte Carlo The Concept Assume that knowledge about a set of nuclear observable parameters employed in nuclear system analyses can be represented by a multi-variate master probability function. This function should be constructed by incorporating the best available information from theory and experiments. The master function is then sampled using Monte Carlo methods to generate a Markov Chain of random observable parameter vectors that ultimately can be employed for a variety of practical applications such as generating evaluations and analyzing the behavior of derived nuclear system parameters.
Master Probability Density Function Bayes Theorem: p(x|T T,E E) = p0(x|T T) L(x|T T,E E) T T signifies prior information based on theory (modeling). E E denotes independent information from experiments that serves to improve (or augment) prior theoretical knowledge T T. x represents random vectors corresponding to possible values of the nuclear observables (e.g., cross sections). The prior probability function p0is based on theory, while likelihoodL is a probability function that quantifies the consistency of data from theory and experiments used to construct the master (posterior) probability function p(x|T T,E E).
Prior Probability Function p0(x|T T) T T is a complicated algorithm that maps theoretical model parametersq to calculatedobservablesx, i.e., x = T T(q). By applying Monte Carlo techniques, a Markov Chain of vectorsqk can be generated by random sampling ofparameters in space S(q) governed by a probability function r0(q). Usually, mean values q0 and covariance matrix Vq are specified. Then, Maximum Entropy suggests r0 should be a normal probability function. A Markov chain of valuesxk in observables space S(x) is generated by Monte Carlo sampling according to xk = T T(qk). The collection {xk} reflects the prior probability function p0, but rarely (if ever) can p0 be expressed explicitly as an analytical function that can be sampled in a conventional way!
Topology Issues S{x} p0(x|T T) and L(x|T T,E E) > 0 so p(x|T T,E E) > 0 The schematic diagram shows mapping from space S{q} to space S{x} by the theoretical (model) algorithmT T. The shaded areas denote regions of non-negligible probability for r0 (green) and p0 (blue). p0(x|T T) > 0 xk L(x|T T,E E) > 0 p0(x|T T) 0 xk = T T(qk) L(x|T T,E E) 0 r0(q) > 0 The region enclosed by a red dashed circle indicates that portion of space S{x} where the likelihood function L(x|T T,E E) is non-negligible. In the region labeled Overlap, where blue and red dashed circles intersect, the master (posterior) function is also non-negligible. S{q} qk r0(q) 0
UMC-G: Analytical Approximation to p0(x|T T) D.L. Smith, Proceedings of AccApp 07, Pocatello, ID, July 29 August 2, 2007, Amer. Nucl. Soc. , p. 736. The collection of K calculated observable parameter vectors {xk} generated by Monte Carlo (see preceding two slides), according to the mapping xk = T T(qk), is used to calculate mean values x0 and covariance matrix Vx via the formulas: x0i ( k=1,K xik) / K and (Vx)ij [( k=1,K xik xjk) / K] - x0i x0j (K is very large). The true prior probability function p0(x|T T) typically is approximated by a multi-variate normal probability function given by: p0(x|T T) C exp {-( )[(x x0)TVx-1 (x x0)]} (C is a normalization constant). Advantage: A lengthy Markov Chain of sample values is thus replaced by an analytical approximation having the same mean values and covariance matrix. This yields a master (posterior) function p(x|T T,E E) that can be sampled readilyby conventional Monte Carlo methods, e.g., Brute Force or Metropolis-Hastings . Disadvantage: This approximation discards all information pertaining to higher- order distribution moments inherent in the Monte Carlo generated Markov chain {xk}. This rejection of information can lead to significant biases in cases where non-linear effects and distribution skewness and kurtosis are present.
UMC-B: Information in p0(x|T T) is Preserved R. Capote et al., Proceedings of ISRD-14, Breton Woods, NH, May 22 27, 2011, ASTM STP-1550, p. 179. The collection of K calculated observable parameter vectors {xk} generated by Monte Carlo (shown in two earlier slides), according to the mapping xk = T T(qk), is preserved. Thus, no information on higher-order moments of p0 is discarded. For each xk, a scalar weighting factor kis generated according to the expression: k = L(xk|T,E). Thus, the worth that is assigned to each MC sampled parameter vector xk is based on its consistency with available experimental data, as reflected in the likelihood function. For very large K, it is assumed that mean values and covariance matrix for the master (posterior) probability function p(x|T T,E E) are estimated from: x0i ( k=1,K kxik) / ( k=1,K k) and (Vx)ij [( k=1,K k xik xjk) / ( k=1,K k)] - x0i x0j The Markov Chain for UMC-B thus consists of the set of pairs {xk, k}. These values can be used for nuclear systems applications as well as evaluations. Advantage: All information in function p0(x|T T) is clearly preserved, including that related to non-linearity as well as the distribution skewness and kurtosis.
A Closer Look at UMC-G and UMC-B Yikes! The region Overlap is very small, probably due to poor agreement between theory and experiments. The areas enclosed by blue and red dashed circles, respectively, indicate regions of non- negligible probability for the model-generated prior probability and the likelihood function that quantifies the consistency of theory and experiment. p0(x|T T) > 0 o o o o The small region Overlap of these two circles is indicative of data inconsistency. Such an outcome could have potentially negative implications for an application of the UMC-B method (e.g., limited or biased sampling of the sparsely sampled region Overlap). Statistical inadequacy is not a problem in applying the UMC-G approach, but it can suffer from significant bias effects due to the explicit rejection of higher-order moments of p0(x|T T). Data inconsistency between theory and experiment will inevitably lead to evaluations and system analysis results that are very questionable and thus inherently unreliable. L(x|T T,E E) > 0 p0(x|T T) and L(x|T T,E E) > 0 so p(x|T T,E E) > 0 Unfinished Business Further investigation of the mentioned sampling issues for the region Overlap in the UMC-B approach is warranted. Detailed inter-comparisons of GLS, UMC- G, and UMC-B predictions for extreme cases and inconsistent data are needed.
UMC-G Plus: A New Option? The original UMC-G formulation discards potentially valuable information about higher moments of the prior probability p0(x|T T) that is reflected in the Markov Chain of vectors {xk} generated by Monte Carlo sampling. It is unlikely that the moments of p0 of higher order than mean values, covariances, skewness, and kurtosis will affect applications significantly. The moments of p0 (mean values, covariances, skewness, and kurtosis) can be estimated using the collection of sample vectors {xk}. Unfinished Business Suggestion: Perhaps analytical functions might be found whose parameters can be adjusted to approximate the distribution moments deduced from {xk}. It could then be employed to serve as a surrogate for the master (posterior) probabilityfunction p(x|T T,E E), and it would then be sampled using conventional Monte Carlo methods. Investigate the structure of realistic MC- generated distributions p0(x|T T) with the intent of quantifying typical mean values, covariances, skewness, and kurtosis. Identify families of analytical mathematical functions that might serve as surrogates for representing the MC-generated distributions p0(x|T T) with greater fidelity than using a simple normal distribution (as in UMC-G).
Thoughts on Likelihood Functions L(x|T T,E E) Available experimental data are usually comprised of mean values and (far less often) covariances. Therefore, comparisons between theoretically calculated observables and experimentalobservables should involve at most mean values and covariances. Consequently, the likelihood function L(x|T T,E E), in accordance with Maximum Entropy, should be an appropriately constructed normal probability function. In particular, it should have the form: L(x|T T,E E) = C exp {-( )[(y yE)TVE-1 (y yE)]} (C is a normalization constant) Note: yE is an experimental data vector with covariance matrix VE. Furthermore, y = f(x), since what is measured (y) may not correspond directly to the observable parameters (x) that are being considered. The function collection f establishes how x and y are related. It may be very difficult to construct a rigorous likelihood function L(x|T T,E E) in any given situation due to one or more of the following limitations: i) incomplete data, ii) discrepant (wrong) data, iii) weak sensitivity relationships between the data (y) and parameters of interest (x), and iv) excessive computational overhead. Unfinished Business Alternatives: Because of these limitations, some investigators (notably A. Koning and D. Rochman) for pragmatic reasons have investigated using simpler alternative likelihood functions L(x|T T,E E). The impact of experimental data quality and availability on applications of UMC needs to be investigated thoroughly. Improve experimental covariance data.
The UMC Approach at a Crossroads? There are unresolved technicalissues and unanswered questions. The way forward to further develop UMC must be clarified. Would UMC-G and UMC-B be truly comparable if p0(x|T T) could be expressed exactly as an analytical function? Can more sophisticated analytical function approximations to a MC prior than the normal distribution be found (e.g., UMC-G Plus)? Are the available experimental data sufficiently accurate and comprehensive to be useful in practice for applying UMC? Can better theoretical models be developed to reduce the discrepancies between theory and quality experimental data? If not, can model-defectsformalisms be developed as practical measures to cope with model vs. experimental data discrepancies? How much extra value does UMCcontribute, compared with GLSQ, to justify the additional effort and computational burden?
