Probabilities, Intervals, What Next? Optimization Problems Related to Extension of Interval Computations to Situations with Partial Information about Probabilities

摘要

When we have only interval ranges [x, x_i] of sample values x_1,...,x_n, what is the interval [V, V] of possible values for the variance V of these values? We show that the problem of computing the upper bound V is NP-hard. We provide a feasible (quadratic time) algorithm for computing the exact lower bound V on the variance of interval data. We also provide feasible algorithms that computes V under reasonable easily verifiable conditions, in particular, in case interval uncertainty is introduced to maintain privacy in a statistical database. We also extend the main formulas of interval arithmetic for different arithmetic operations x_1, op x_2 to the case when, for each input x_i, in addition to the interval x_i= [x_i, x_i] of possible values, we also know its mean E_i (or an interval E_i of possible values of the mean), and we want to find the corresponding bounds for y=x_1 op x_2 and its mean. In this case, we are interested not only in the bounds for y. but also in the bounds for the mean of y. We formulate and solve the corresponding optimization problems, and describe remaining open problems.