Uncertainty and Imprecision

Abstract

The core problem in the theory of measurements is to assign a value to represent quantitatively our knowledge of a physical property, given a finite set of observations. A corollary problem is to provide a statement as to the quality of the measurement. It is asserted that measurement is a process of drawing plausible inferences from incomplete data. A procedure for treating measurement processes is developed following Jaynes’ formalism, wherein probability is interpreted subjectively as a state of knowledge. In addition to the set of possible outcomes, it is necessary to include prior knowledge. In this paper, only stationary measurement processes are considered, employing repetitive observations of the same quantity. The set of possible observations is exhaustive and mutually exclusive. A theory that predicts infinite deviations cannot be accepted. Three conclusions are presented: (a) When only the expectation value and the variance are assumed known, the least biased, probability distribution is Gaussian; (b) when consideration of a possible malfunction is included, the probability distribution is bounded at three or possibly four standard deviations; (c) when the estimate of the expectation value is studied for a single set of observations, the probability distribution reduces to a rectangular function between the specified bounds. Uncertainty is defined by Shannon’s theorem. Imprecision is defined as the bounds on the probability distribution.