Measured Averages and Inferred Extremes: Info-Gap Analysis of Deep Uncertainty

Abstract

Averages are measured for diagnosis, prediction, or surveillance. However, averages reveal nothing about fluctuations, and extreme values may be more significant than the average. The analyst can choose decision variables: path length and other parameters. This paper explores the choice of decision variables to achieve robustness against pernicious uncertainty when interpreting an average, in face of uncertain fluctuations of the averaged variable. We also explore the choice of decision variables to achieve opportuneness from propitious uncertainty. Trade-offs and “trade-ons” between robust and opportune decision variables are identified. Three examples are developed: enforcing speed limits; inferring levels of economic activity; and statistical hypothesis testing. We use concepts of robustness and opportuneness from info-gap decision theory. We also explore the relation between the probability of success and the non-probabilistic robustness to uncertainty, demonstrating conditions where robustness is a proxy for probability.

A Maximum Acceleration

Consider any infinitesimal segment of length $\mathrm{d}x$ along the road. The speeds at the start and end of this segment are related as:

$$\begin{aligned} v(x + \mathrm{d}x) = v(x) + \frac{\mathrm{d}v(x)}{\mathrm{d}x} \mathrm{d}x. \end{aligned}$$

(74)

The derivative in this relation can be written:

$$\begin{aligned} \frac{\mathrm{d}v(x)}{\mathrm{d}x} = \frac{\mathrm{d}v(t)}{\mathrm{d}t} \frac{\mathrm{d}t}{\mathrm{d}x} = \frac{\dot{v}(t)}{v(x)}, \end{aligned}$$

(75)

where $\dot{v}(t)$ is the temporal acceleration of the car. Combining the last two relations yields:

$$\begin{aligned} v(x + \mathrm{d}x) = v(x) + \frac{\dot{v}(t)}{v(x)} \mathrm{d}x. \end{aligned}$$

(76)

Hence, assuming that v(x) is positive, the increment in velocity on any infinitesimal segment of road is maximal if the car accelerates maximally. The cumulative effect is that the final speed is maximal if the car accelerates maximally throughout the travel, even though this actually minimizes the time during which acceleration occurs.