From Interval Methods of Representing Uncertainty to a General Description of Uncertainty Traditional Description of Uncertainty in Science and Engineering and Its Drawbacks Deenition 2. for Every Two Probability Distributions

Measurements do not result in an exact value of the measured quantity; even after the most accurate measurement , there is still some uncertainty about the actual value of the measured quantity. Traditionally, in science and engineering, this uncertainty is characterized by a probability distribution; however, often, we do not know this probability distribution exactly. So, to get a more adequate description of this uncertainty, we must consider classes of possible probability distributions. A natural question is: Are all possible classes needed for this description? In this paper, we show that even for simple situations, we indeed need arbitrary closed convex classes of probability distributions. Uncertainty is typical in science and engineering. A large portion of knowledge in science and engineering comes from measurements. Some of this knowledge comes not from measurements but from the expertise of scientists and engineers; however, to make sure that this additional knowledge is indeed correct, we must check it against the results of the measurements. In short, measurements are the basis of modern science and engineering. By deenition, a measurement means measuring the value of a physical quantity. Ideally, we would like to get the exact value of this quantity, but in real life, measurements are never 100% accurate. Due to inevitable noise, inaccuracies, etc., the results of the measurement are never absolutely accurate. After the measurement, we do not get the exact value of the measured quantity , because several close values are consistent with the same measurement result. Therefore, when we process measurement results in science and engineering, we must take the measurement uncertainty into consideration. Traditional approach to describing measurement uncertainty. To characterize the measurement uncertainty , we must know: which exactly values of the measured quantity are possible (provided the given measurement results), and which of these possible values are more probable. In accordance with this idea, traditionally in science and engineering, uncertainty of the measurement result is characterized by describing the set of all possible values of the measured quantity, and in describing, for each possible value of the measured quantity, the probability that this value is the actual one. This probability can be viewed as a frequency of this particular value among all experiments in which we get the given measurement results. Let us make this intuitive description more mathematically accurate. For continuous measured quantities , the probability of each exact value is usually 0, we can only …