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G. Filippi and M. Vasile
the surrogate was periodically updated to identify the threshold values where the
approximation was the closest to the true Bel. These two improvements allow one
to insert selected values of the surrogate in the optimisation loop and improve the
Bel at a discrete number of thresholds ν.
If sampling is used to build a surrogate of the quantity of interest, the computational cost due to the maximisation over all the focal elements is only
partially mitigated, as the cost of each optimisation is reduced but the number of
optimisations might remain very high and could still scale exponentially with the
number of dimensions of the uncertainty space U .
17.4.2 Dimensionality Reduction
Another class of approaches tries to reduce the number of focal elements upfront.
The general idea is to define criteria to sort the FEs by their importance and then
approximate the m-function in Eq. (17.15) to a m -function with a lower number of
FEs. A few examples follow:
• The Bayesian approximation proposed by Voorbraak’s [83] produces a discrete
probability distribution: the new mass function m considers only the singleton
subsets θ i in the Power Set 2 Θ .
• Dubois and Prade’s consonant approximations [84]
• The k-l-x method proposed by Tessem [85] uses the bpa’s as sorting criterion.
The approximating m =m klx then includes only the p FEs with higher bpa,
where k < p < l, and such that the sum of the masses of the deleted F Es is
less than x. A normalisation method is finally used to redistribute the total mass
of the deleted F Es to the remaining ones.
• The summarisation method takes the first p focal elements with the highest bpa,
as in [85], and lumps together all the remaining ones in a single focal element
with a bpa that is the sum of their bpa’s .
• The D1 method [86] beside the criteria of mass introduces also the cardinality.
• The Batch approximation method and Iterative approximation method [87]
suggest, instead, that mass and cardinality are not sufficient to discriminate
which F E to take and which one to discard. Then in the paper a non-redundancy
measure is presented based on the definition of distance between two F Es as
proposed by Denœux in [88].
• for some heuristic methods refer to [89].
17.4.3 Outer Belief Estimation via Evolutionary Binary Tree
It is generally desirable to have a method that produces estimated values of Bel that
are lower than the actual one. We call these estimated values: outer approximations.
G. Filippi and M. Vasile
the surrogate was periodically updated to identify the threshold values where the
approximation was the closest to the true Bel. These two improvements allow one
to insert selected values of the surrogate in the optimisation loop and improve the
Bel at a discrete number of thresholds ν.
If sampling is used to build a surrogate of the quantity of interest, the computational cost due to the maximisation over all the focal elements is only
partially mitigated, as the cost of each optimisation is reduced but the number of
optimisations might remain very high and could still scale exponentially with the
number of dimensions of the uncertainty space U .
17.4.2 Dimensionality Reduction
Another class of approaches tries to reduce the number of focal elements upfront.
The general idea is to define criteria to sort the FEs by their importance and then
approximate the m-function in Eq. (17.15) to a m -function with a lower number of
FEs. A few examples follow:
• The Bayesian approximation proposed by Voorbraak’s [83] produces a discrete
probability distribution: the new mass function m considers only the singleton
subsets θ i in the Power Set 2 Θ .
• Dubois and Prade’s consonant approximations [84]
• The k-l-x method proposed by Tessem [85] uses the bpa’s as sorting criterion.
The approximating m =m klx then includes only the p FEs with higher bpa,
where k < p < l, and such that the sum of the masses of the deleted F Es is
less than x. A normalisation method is finally used to redistribute the total mass
of the deleted F Es to the remaining ones.
• The summarisation method takes the first p focal elements with the highest bpa,
as in [85], and lumps together all the remaining ones in a single focal element
with a bpa that is the sum of their bpa’s .
• The D1 method [86] beside the criteria of mass introduces also the cardinality.
• The Batch approximation method and Iterative approximation method [87]
suggest, instead, that mass and cardinality are not sufficient to discriminate
which F E to take and which one to discard. Then in the paper a non-redundancy
measure is presented based on the definition of distance between two F Es as
proposed by Denœux in [88].
• for some heuristic methods refer to [89].
17.4.3 Outer Belief Estimation via Evolutionary Binary Tree
It is generally desirable to have a method that produces estimated values of Bel that
are lower than the actual one. We call these estimated values: outer approximations.