Uncertainty bracket notation

Uncertainty is a situation which involves imperfect and/or unknown information. However, "uncertainty is an unintelligible expression without a straightforward description". It arises in subtly different ways in a number of fields, including insurance, philosophy, physics, statistics, economics, finance, psychology, sociology, engineering, metrology, and information science. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable and/or environments, as well as due to ignorance and/or indolence.

Although the terms are used in various ways among the general public, many specialists in decision theory, statistics and other quantitative fields have defined uncertainty, risk, and their measurement as:

In economics, Frank Knight distinguished risk and uncertainty; uncertainty being risk that is immeasurable, not possible to calculate, and referred to as Knightian uncertainty:

Uncertainty must be taken in a sense radically distinct from the familiar notion of risk, from which it has never been properly separated.... The essential fact is that 'risk' means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating.... It will appear that a measurable uncertainty, or 'risk' proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all.

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Wikipedia