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D. Krpelík and T. Basu
2.4.5 Probability Boxes
A probability box (P-box) is defined by two cumulative distribution functions F ≥
F [20]. The set of probability distributions, which a P-box envelopes, is composed
of all the CDFs which are bounded by F and F . The resulting lower probability P
is ∞-monotone and coherent. In the case that an additional knowledge is available,
additional bounds may also be imposed, e.g. on the mean values and variances of
the enveloped distributions to further narrow the structure. Standard arithmetical
operations can be generalised for P-boxes so they can be used for risk and sensitivity
analysis, similarly to the random sets.
In the multivariate case, multiple assumptions about the correlation of RVs can
be imposed, including the situation when the correlation is regarded as entirely
unknown [21]. An example of inferences from different dependency assumptions
is shown in Fig. 2.5 for the addition of two RVs with a uniform distribution (any
precise distribution is also a P-box with a single element in its credal set). So, Pboxes allow us to refrain entirely from specifying any assumption about dependency.
The calculation is based on the Frechet inequalities [21].
Uniform distribution U(0,1)
1
1
0.5
0.5
0
0
0
0
0.5
0.5
1
1
x
x
Frechet Dependency
1
1
0.5
0.5
0
0
0
0
0.5
0.5
0.5
0.5
1
1
2
2
x
x
CDF
CDF
CDF
CDF
Uniform distribution U(0,1)
Independence assumption
Fig. 2.5 P-boxes for the sum of two uniform RVs. P-boxes for the uniform RVs are shown at
the top portion, whereas the P-boxes of their sum based on different dependency assumptions are
presented at the bottom part: no assumption (left) and independence assumption (right)
D. Krpelík and T. Basu
2.4.5 Probability Boxes
A probability box (P-box) is defined by two cumulative distribution functions F ≥
F [20]. The set of probability distributions, which a P-box envelopes, is composed
of all the CDFs which are bounded by F and F . The resulting lower probability P
is ∞-monotone and coherent. In the case that an additional knowledge is available,
additional bounds may also be imposed, e.g. on the mean values and variances of
the enveloped distributions to further narrow the structure. Standard arithmetical
operations can be generalised for P-boxes so they can be used for risk and sensitivity
analysis, similarly to the random sets.
In the multivariate case, multiple assumptions about the correlation of RVs can
be imposed, including the situation when the correlation is regarded as entirely
unknown [21]. An example of inferences from different dependency assumptions
is shown in Fig. 2.5 for the addition of two RVs with a uniform distribution (any
precise distribution is also a P-box with a single element in its credal set). So, Pboxes allow us to refrain entirely from specifying any assumption about dependency.
The calculation is based on the Frechet inequalities [21].
Uniform distribution U(0,1)
1
1
0.5
0.5
0
0
0
0
0.5
0.5
1
1
x
x
Frechet Dependency
1
1
0.5
0.5
0
0
0
0
0.5
0.5
0.5
0.5
1
1
2
2
x
x
CDF
CDF
CDF
CDF
Uniform distribution U(0,1)
Independence assumption
Fig. 2.5 P-boxes for the sum of two uniform RVs. P-boxes for the uniform RVs are shown at
the top portion, whereas the P-boxes of their sum based on different dependency assumptions are
presented at the bottom part: no assumption (left) and independence assumption (right)