Intervals and probability bounds

Ferson (1996) points out that there is a class of problems in which information may be known regarding upper and lower bounds for a particular variable, but not regarding a probability distribution. The model output in such cases is an interval, rather than a distribution. Such problems are more appropriately dealt with using interval methods rather than imposing probabilistic assumptions upon each input. Interval methods can be extended to situations in which marginal probability distributions are specified for each model input but for which the dependence between the distributions is not known. Thus, rather than assume statistical independence, or any particular correlation or more complex form of dependency, a bounding technique can be used to specify the range within which the model output distribution must be bounded. 

Some would argue that if there is sufficient information upon which to quantify ranges, then there is also likely information upon which to base a judgement regarding the type of distribution that could be used to describe uncertainty in the input. Interval methods, however, may be useful for problems in which there may be complex dependencies between inputs but for which the dependencies are not known. A disadvantage of interval methods is that the predicted intervals for a model output can be quite wide, since they are informed only by the end-points of the ranges for each input. Interval methods can be useful as a quality 

As a simple illustration of interval methods, consider the example given by Ferson (1996) pertaining to multiplication of two inputs. Input A has an interval of [0.2, 0.4], and Input B has an interval of [0.3, 0.5]. The interval for the model output is [0.06, 0.2]. The output interval is the narrowest possible interval that accounts for all possible forms of dependence between A and B. 

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