Adding Constraints to Situations When, In Addition to Intervals,We Also Have Partial Information about Probabilities

In many practical situations, we need to combine probabilistic and interval uncertainty. For example, we need to 1 A compute statistics like population mean E = 1/n.nSigmai=1xi or population variance V = 1/nnSigmai=1(xi-E)2 in situations when we only know intervals xi of possible value of xi. In this case, it is desirable to compute the range of the corresponding characteristic. Some range computation problems are NP-hard; for these problems, in general, only an enclosure is possible. For other problems, there are efficient algorithms. In many practical situations, we have additional information that can be used as constraints on possible cumulative distribution functions (cdfs). For example, we may know that the actual (unknown) cdf is Gaussian. In this paper, we show that such constraints enable us to drastically narrow down the resulting ranges - and sometimes, transform the originally intractable (NP-hard) computational problem of computing the exact range into an efficiently solvable one. This possibility is illustrated on the simplest example of an NP-problem from interval statistics: the problem of computing the range V of the variance V. We also describe how we can estimate the amount of information under such combined intervals-and-constraints uncertainty.