Uncertainty is a term used in subtly different ways in a number of fields, including philosophy, statistics, economics,
finance, insurance, psychology, engineering and science. It
applies to predictions of future events, to physical measurements already made, or to the
unknown unknown.
Relation between uncertainty, probability, vagueness and risk
In his seminal work Risk, Uncertainty, and Profit[1] University of Chicago economist Frank Knight (1921) established the important distinction between risk and
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."
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 and risk more specifically. Doug Hubbard defines uncertainty and risk as:[2]
-
- Uncertainty: The lack of certainty, A state of having limited knowledge where it is impossible to exactly describe
existing state or future outcome, more than one possible outcome.
- Measurement of Uncertainty:A set of possible states or outcomes where probabilities are assigned to each possible
state or outcome - this also includes the application of a probability density function to continuous variables
- Risk:A state of uncertainty where some possible outcomes have an undesired effect or significant loss.
- Measurement of Risk:A set of measured uncertainties where some possible outcomes are losses, and the magnitudes of
those losses - this also includes loss functions over continuous variables.
There are other different taxonomy of uncertainties and decisions that include a more broad sense of uncertainty and how it
should be approached from an ethics perspective [3]:
For example, if you do not know whether it will rain tomorrow, then you have a state of uncertainty. If you apply probabilities
to the possible outcomes using weather forecasts or even just a calibrated
probability assessment, you have quantified the uncertainty. Suppose you quantify your uncertainty as a 90% chance of
sunshine. If you are planning a major, costly, outdoor event for tomorrow then you have risk since there is a 10% chance of rain
and rain would be undesirable. Furthermore, if this is a business event and you would lose $100,000 if it rains, then you have
quantified the risk (a 10% chance of losing $100,000). These situation can be made even more realistic by quantifying light rain
vs. heavy rain, the cost of delays vs. outright cancellation, etc.
Some may represent the risk in this example as the "expected opportunity loss" (EOL) or the chance of the loss multiplied by
the amount of the loss (10% x $100,000 = $10,000). That is useful if the organizer of the event is "risk neutral" which most
people are not. Most would be willing to pay a premium to avoid the loss. An insurance
company, for example, would compute an EOL as a minimum for any insurance coverage, then add on to that other operating costs and
profit. Since many people are willing buy insurance for many reasons, then clearly the EOL alone is not the perceived value of
avoiding the risk.
Quantitative uses of the terms uncertainty and risk are fairly consistent from fields such as probability theory, actuarial science, and
information theory. Some also create new terms without substantially changing the
definitions of uncertainty or risk. For example, surprisal is a variation on
uncertainty sometimes uses in information theory. But outside of the more
mathematical uses of the term, usage may vary widely. In cognitive psychology,
uncertainty can be real, or just a matter of perception, such as expectations, threats,
etc.
Vagueness or ambiguity are sometimes described as "second order uncertainty", where there is uncertainty even about the
definitions of uncertain states or outcomes. The difference here is that this uncertainty is about the human definitions and
concepts not an objective fact of nature. It has been argued that ambiguity, however, is always avoidable while uncertainty (of
the "first order" kind) is not necessarily avoidable.[4]:
Uncertainty may be purely a consequence of a lack of knowledge of obtainable facts. That is, you may be uncertain about
whether a new rocket design will work, but this uncertainty can be removed with further analysis and experimentation. At the
subatomic level, however, uncertainty may be a fundamental and unavoidable property of the universe. In quantum mechanics, the Heisenberg Uncertainty Principle
puts limits on how much an observer can ever know about the position and velocity of a particle. This may not just be ignorance
of potentially obtainable facts but that there is no fact to be found. There is some controversy in physics as to whether such
uncertainty is an irreducible property of nature or if there are "hidden variables" that would describe the state of a particle
even more exactly that Heisenberg's uncertainty principle allows.
Relation between uncertainty, accuracy, precision, standard deviation, standard error, and confidence interval
The uncertainty of a measurement is stated by giving a range of values which are likely to enclose the true value. This may be
denoted by error bars on a graph, or as value ± uncertainty, or as decimal
fraction(uncertainty). The latter "concise notation" is used for example by IUPAC in stating the atomic mass of elements. There, 1.00794(7)
stands for 1.00794 ± 0.00007.
Often, the uncertainty of a measurement is found by repeating the measurement enough times to get a good estimate of the
standard deviation of the values. Then, any single value has an uncertainty equal to
the standard deviation. However, if the values are averaged and the mean is reported, then the averaged measurement has
uncertainty equal to the standard error which is the standard deviation
divided by the square root of the number of measurements.
When the uncertainty represents the standard error of the measurement, then about 68.2% of the time, the true value of the
measured quantity falls within the stated uncertainty range. For example, it is likely that for 31.8% of the atomic mass values
given on the list of elements by atomic mass, the true value lies
outside of the stated range. If the width of the interval is doubled, then probably only 4.6% of the true values lie outside the
doubled interval, and if the width is tripled, probably only 0.3% lie outside. These values follow from the properties of the
normal distribution, and they apply only if the measurement process produces
normally distributed errors. In that case, the quoted standard errors are
easily converted to 68.2% ("one sigma"), 95.4% ("two sigma"), or 99.7% ("three sigma") confidence intervals.
Fields of activities or knowledge where uncertainty is important
- Investing in financial markets such as the stock market.
- Uncertainty is used in engineering notation when talking about significant
figures. Or the possible error involved in measuring things such as distance.
- Uncertainty is designed into games, most notably in gambling,
where chance is central to play.
- In scientific modelling, in which the prediction of future events should be
understood to have a range of expected values.
- In physics in certain situations, uncertainty has been elevated into a principle, the
uncertainty principle.
- In weather forecasting it is now commonplace to include data on the degree of
uncertainty in a weather forecast.
- Uncertainty is often an important factor in economics. According to economist
Frank Knight, it is different from risk, where there is a
specific probability assigned to each outcome (as when flipping a fair coin). Uncertainty
involves a situation that has unknown probabilities, while the estimated probabilities of possible outcomes need not add to
unity.
- In metrology, measurement uncertainty is
a central concept quantifying the dispersion one may reasonably attribute to a measurement result. Such an uncertainty can also
be referred to as a measurement error. In daily life, measurement uncertainty is often implicit
("He is 6 feet tall" give or take a few inches), while for any serious use an explicit statement of the measurement uncertainty
is necessary. The expected measurement uncertainty of many measuring instruments
(scales, oscilloscopes, force gages, rulers, thermometers, etc) is often stated in the manufacturers specification.
- The most commonly used procedure for calculating measurement uncertainty is described in the Guide to the Expression of
Uncertainty in Measurement (often referred to as "the GUM") published by ISO. A derived work is for example the National
Institute for Standards and Technology (NIST) publication
NIST Technical Note 1297 "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results" and the
Eurachem/Citac publication "Uncertatinty in measurements" (available at the Eurachem homepage). The uncertainty of the result of
a measurement generally consists of several components. The components are regarded as random
variables, and may be grouped into two categories according to the method used to estimate their numerical values:
-
- Type A, those which are evaluated by statistical methods,
- Type B, those which are evaluated by other means, e.g. by assigning a probability distribution.
- By propagating the variances of the components through a function relating the components
to the measurement result, the combined measurement uncertainty is given as the square root of the resulting variance. The
simplest form is the standard deviation of a repeated observation.
Uncertainty as an artistic theme
Uncertainty has been a common theme in art, both as a thematic device (see, for example, the indecision of Hamlet), and as a quandary for the artist (such as Martin Creed's
difficulty with deciding what artworks to make).
See also
References
- ^ Knight, F.H. (1921) Risk, Uncertainty, and Profit. Boston, MA: Hart,
Schaffner & Marx; Houghton Mifflin Company
- ^ Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles
in Business", John Wiley & Sons, 2007
- ^ Tannert C, Elvers HD, Jandrig B
(2007). "The ethics of uncertainty. In the light of possible dangers, research becomes a moral duty.". EMBO Rep. 8
(10): 892-6. DOI:10.1038/sj.embor.7401072. PMID 17906667.
- ^ Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles
in Business", John Wiley & Sons, 2007
External links
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