This bounding approach permits analysts to make calculations without requiring overly precise assumptions about parameter values, dependence among variables, or even distribution shape.
It also gives the same answers as Monte Carlo simulation does when information is abundant enough to precisely specify input distributions and their dependencies.
In some cases, a calculated p-box will also be best-possible in the sense that the bounds could be no tighter without excluding some of the possible distributions.
This property often suffices for use in risk analysis and other fields requiring calculations under uncertainty.
Indeed, in 1854 George Boole used the notion of interval bounds on probability in his The Laws of Thought.
Of particular note is Fréchet's derivation in the 1930s of bounds on calculations involving total probabilities without dependence assumptions.
[4]) The methods of probability bounds analysis that could be routinely used in risk assessments were developed in the 1980s.
Hailperin[2] described a computational scheme for bounding logical calculations extending the ideas of Boole.
Yager[5] described the elementary procedures by which bounds on convolutions can be computed under an assumption of independence.
Since that time, formulas and algorithms for sums have been generalized and extended to differences, products, quotients and other binary and unary functions under various dependence assumptions.
[9][10][11][12][13][14] Arithmetic expressions involving operations such as additions, subtractions, multiplications, divisions, minima, maxima, powers, exponentials, logarithms, square roots, absolute values, etc., are commonly used in risk analyses and uncertainty modeling.
There are convenient algorithms for computing these generalized convolutions under a variety of assumptions about the dependencies among the inputs.
is a random variable governed by the distribution function F, that is, Let us generalize the tilde notation for use with p-boxes.
[13] The convolution under the intermediate assumption that X and Y have positive dependence is likewise easy to compute, as is the convolution under the extreme assumptions of perfect positive or perfect negative dependency between X and Y.
If the probabilities of events are characterized by intervals, as suggested by Boole[1] and Keynes[3] among others, these binary operations are straightforward to evaluate.
In this case, we can infer at least that the probability of the joint event A & B is surely within the interval where env([x1,x2], [y1,y2]) is [min(x1,y1), max(x2,y2)].
Some analysts[15][16][17][18][19][20] use sampling-based approaches to computing probability bounds, including Monte Carlo simulation, Latin hypercube methods or importance sampling.
Thus, unlike the analytical theorems or methods based on mathematical programming, sampling-based calculations usually cannot produce verified computations.
However, sampling-based methods can be very useful in addressing a variety of problems which are computationally difficult to solve analytically or even to rigorously bound.
One important example is the use of Cauchy-deviate sampling to avoid the curse of dimensionality in propagating interval uncertainty through high-dimensional problems.
[21] PBA belongs to a class of methods that use imprecise probabilities to simultaneously represent aleatoric and epistemic uncertainties.
PBA is a generalization of both interval analysis and probabilistic convolution such as is commonly implemented with Monte Carlo simulation.
P-boxes and probability bounds analysis have been used in many applications spanning many disciplines in engineering and environmental science, including: