A probability box (or p-box) is a characterization of an uncertain number consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed. Probability bounds analysis is used to make arithmetic and logical calculations with p-boxes. An example p-box is shown in the figure at right for an uncertain number x consisting of a left (upper) bound and a right (lower) bound on the probability distribution for x. The bounds are coincident for values of x below 0 and above 24. The bounds may have almost any shape, including step functions, so long as they are monotonically increasing and do not cross each other. A p-box is used to express simultaneously incertitude (epistemic uncertainty), which is represented by the breadth between the left and right edges of the p-box, and variability (aleatory uncertainty), which is represented by the overall slant of the p-box.
Interpretation
There are dual interpretations of a p-box. It can be understood as bounds on the cumulative probability associated with any x-value. For instance, in the p-box depicted at right, the probability that the value will be 2.5 or less is between 4% and 36%. A p-box can also be understood as bounds on the x-value at any particular probability level. In the example, the 95th percentile is sure to be between 9 and 16. If the left and right bounds of a p-box are sure to enclose the unknown distribution, the bounds are said to be rigorous, or absolute. The bounds may also be the tightest possible such bounds on the distribution function given the available information about it, in which case the bounds are therefore said to be best-possible. It may commonly be the case, however, that not every distribution that lies within these bounds is a possible distribution for the uncertain number, even when the bounds are rigorous and best-possible.
Mathematical definition P-boxes are specified by left and right bounds on the distribution function (or, equivalently, the survival function) of a quantity and, optionally, additional information constraining the quantity's mean and variance to specified intervals, and specified constraints on its distributional shape (family, unimodality, symmetry, etc.). A p-box represents a class of probability distributions consistent with these constraints. A distribution function on the real numbers R {\displaystyle \mathbb {R} } , is a function D : R → [ 0 , 1 ] , {\displaystyle D:\mathbb {R} \rightarrow [0,1],} for which D(x) ≤ D(y) whenever x < y, and the limit of D at +∞ is 1 and the limit at −∞ is 0. A p-box is a set of distributions functions F satisfying the following constraints, for specified distribution functions F F, and specified bounds m1 ≤ m2 on the expected value of the distribution and specified bounds v1 ≤ v2 on the variance of the distribution.
F _ ( x ) ≤ F ( x ) ≤ F ¯ ( x ) , m 1 ≤ ∫ − ∞ ∞ x d F ( x ) ≤ m 2 v 1 ≤ ∫ − ∞ ∞ x 2 d F ( x ) − ( ∫ − ∞ ∞ x d F ( x ) ) 2 ≤ v 2 F ∈ F {\displaystyle {\begin{aligned}&{\underline {F}}(x)\leq F(x)\leq {\overline {F}}(x),\\[4pt]&m_{1}\leq \int _{-\infty }^{\infty }x\,\mathrm {d} F(x)\leq m_{2}\\[4pt]&v_{1}\leq \int _{-\infty }^{\infty }x^{2}\,\mathrm {d} F(x)-\left(\int _{-\infty }^{\infty }x\,\mathrm {d} F(x)\right)^{2}\leq v_{2}\\[4pt]&F\in \mathbf {F} \end{aligned}}}
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