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 , is a function for which whenever , and the limit of 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, and specified bounds m1 ≤ m2 on the expected value of the distribution and specified bounds v1 ≤ v2 on the variance of the distribution.
where integrals of the form are Riemann–Stieltjes integrals.
Thus, the constraints are that the distribution function F falls within prescribed bounds, the mean of the distribution is in the interval m, the variance of the distribution is in the interval v, and the distribution is within some admissible class of distributions F. The Riemann–Stieltjes integrals do not depend on the differentiability of F.
P-boxes serve the same role for random variables that upper and lower probabilities serve for events. In robust Bayes analysisBerger, J. O. (1984). "The robust Bayesian viewpoint." Pages 63–144 in Robustness of Bayesian analyses, edited by J.B. Kadane, Elsevier Science. a p-box is also known as a distribution band.Basu, S. (1994). "Variations of posterior expectations for symmetric unimodal priors in a distribution band." Sankhyā: The Indian Journal of Statistics, Series A 56: 320–334.Basu, S., and A. DasGupta (1995). "Robust Bayesian analysis with distribution bands". Statistics and Decisions 13: 333–349. A p-box can be constructed as a closed neighborhood of a distribution under the Kolmogorov, Lévy or Wasserstein metric. A p-box is a crude but computationally convenient kind of credal set. Whereas a credal set is defined solely in terms of the constraint F as a convex set of distributions (which automatically determine , F, m, and v, but are often very difficult to compute with), a p-box usually has a loosely constraining specification of F, or even no constraint so that . Calculations with p-boxes, unlike credal sets, are often quite efficient, and algorithms for all standard mathematical functions are known.
A p-box is minimally specified by its left and right bounds, in which case the other constraints are understood to be vacuous as Even when these ancillary constraints are vacuous, there may still be nontrivial bounds on the mean and variance that can be inferred from the left and right edges of the p-box.
Where p-boxes come from P-boxes may arise from a variety of kinds of incomplete information about a quantity, and there are several ways to obtain p-boxes from data and analytical judgment.
Distributional p-boxes When a probability distribution is known to have a particular shape (e.g., normal, uniform, beta, Weibull, etc.) but its parameters can only be specified imprecisely as intervals, the result is called a distributional p-box, or sometimes a parametric p-box. Such a p-box is usually easy to obtain by enveloping extreme distributions given the possible parameters. For instance, if a quantity is known to be normal with mean somewhere in the interval [7,8] and standard deviation within the interval [1,2], the left and right edges of the p-box can be found by enveloping the distribution functions of four probability distributions, namely, normal(7,1), normal(8,1), normal(7,2), and normal(8,2), where normal(μ,σ) represents a normal distribution with mean μ and standard deviation σ. All probability distributions that are normal and have means and standard deviations inside these respective intervals will have distribution functions that fall entirely within this p-box. The left and right bounds enclose many non-normal distributions, but these would be excluded from the p-box by specifying normality as the distribution family.
Distribution-free p-boxes Even if the parameters such as mean and variance of a distribution are known precisely, the distribution cannot be specified precisely if the distribution family is unknown. In such situations, envelopes of all distributions matching given moments can be constructed from inequalities such as those due to Markov, Chebyshev, Cantelli, or RoweRowe, N.C. (1988). Absolute bounds on the mean and standard deviation of transformed data for constant-sign-derivative transformations. SIAM Journal of Scientific and Statistical Computing 9: 1098–1113.Smith, J.E. (1995). Generalized Chebychev inequalities: theory and applications in decision analysis. Operations Research 43: 807–825. that enclose all distribution functions having specified parameters. These define distribution-free p-boxes because they make no assumption whatever about the family or shape of the uncertain distribution. When qualitative information is available, such as that the distribution is unimodal, the p-boxes can often be tightened substantially.Zhang, J. and D. Berleant (2005). Arithmetic on random variables: squeezing the envelopes with new joint distribution constraints. Pages 416–422 in Proceedings of the Fourth International Symposium On Imprecise Probabilities and Their Applications (ISIPTA '05), Carnegie Mellon University, Pittsburgh, July 20–23, 2005.
P-boxes from imprecise measurements When all members of a population can be measured, or when random sample data are abundant, analysts often use an empirical distribution to summarize the values. When those data have non-negligible measurement uncertainty represented by interval ranges about each sample value, an empirical distribution may be generalized to a p-box.Ferson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg (2007). Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty. Sandia National Laboratories, SAND 2007-0939, Albuquerque, NM. Such a p-box can be specified by cumulating the lower endpoints of all the interval measurements into a cumulative distribution forming the left edge of the p-box, and cumulating the upper endpoints to form the right edge. The broader the measurement uncertainty, the wider the resulting p-box.
Interval measurements can also be used to generalize distributional estimates based on the method of matching moments or maximum likelihood, that make shape assumptions such as normality or lognormality, etc.Xiang, G., V. Kreinovich and S. Ferson, (2007). Fitting a normal distribution to interval and fuzzy data. Pages 560–565 in Proceedings of the 26th International Conference of the North American Fuzzy Information Processing Society NAFIPS'2007, M. Reformat and M.R. Berthold (eds.). Although the measurement uncertainty can be treated rigorously, the resulting distributional p-box generally will not be rigorous when it is a sample estimate based on only a subsample of the possible values. But, because these calculations take account of the dependence between the parameters of the distribution, they will often yield tighter p-boxes than could be obtained by treating the interval estimates of the parameters as unrelated as is done for distributional p-boxes.
Confidence bands There may be uncertainty about the shape of a probability distribution because the sample size of the empirical data characterizing it is small. Several methods in traditional statistics have been proposed to account for this sampling uncertainty about the distribution shape, including Kolmogorov–SmirnovKolmogorov, A. (1941). Confidence Limits for an Unknown Distribution Function. Annals of Mathematical Statistics 12: 461–463. and similarOwen, A.B. (1995). Nonparametric likelihood confidence bands for a distribution function. Journal of the American Statistical Association 90: 516–521. confidence bands, which are distribution-free in the sense that they make no assumption about the shape of the underlying distribution. There are related confidence-band methods that do make assumptions about the shape or family of the underlying distribution, which can often result in tighter confidence bands.Cheng, R.C.H., and T.C. Iles (1983). Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25: 77–86.Cheng, R.C.H., B.E. Evans and J.E. Williams (1988). Confidence band estimations for distributions used in probabilistic design. International Journal of Mechanical Sciences 30: 835–845.Murphy, S.A. (1995). Likelihood ratio-based confidence intervals in survival analysis. Journal of the American Statistical Association 90: 1399–1405. Constructing confidence bands requires one to select the probability defining the confidence level, which usually must be less than 100% for the result to be non-vacuous. Confidence bands at the (1 − α)% confidence level are defined such that, (1 − α)% of the time they are constructed, they will completely enclose the distribution from which the data were randomly sampled. A confidence band about a distribution function is sometimes used as a p-box even though it represents statistical rather than rigorous or sure bounds. This use implicitly assumes that the true distribution, whatever it is, is inside the p-box.