Reference class problem

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In statistics, the reference class problem is the problem of deciding what class to use when calculating the probability applicable to a particular case. For example, to estimate the probability of an aircraft crashing, one might use the frequency of crashes of all aircraft, of this make of aircraft, of aircraft flown by this company in the last ten years, etc. Any case is a member of very many classes, in which the frequency of the attribute of interest (such as crashing) differs, and the reference class problem discusses which is the most appropriate to use.

More formally, many arguments in statistics take the form of a statistical syllogism:

1. $X$ proportion of $F$ are $G$
2. $I$ is an $F$
3. Therefore, the chance that $I$ is a $G$ is $X$

$F$ is called the "reference class" and $G$ is the "attribute class" and $I$ is the individual object. How is one to choose an appropriate class $F$?

In Bayesian statistics, the problem arises at that of deciding on a prior probability for the outcome in question (or when considering multiple outcomes, a prior probability distribution).

History

John Venn stated in 1876 that "every single thing or event has an indefinite number of properties or attributes observable in it, and might therefore be considered as belonging to an indefinite number of different classes of things", leading to problems with how to assign probabilities to a single case. He used as an example the probability that John Smith, a consumptive Englishman aged fifty, will live to sixty-one.

The name "problem of the reference class" was given by Hans Reichenbach, who wrote, "If we are asked to find the probability holding for an individual future event, we must first incorporate the event into a suitable reference class. An individual thing or event may be incorporated in many reference classes, from which different probabilities will result."

There has also been discussion of the reference class problem in philosophy.

See also

• Statistical syllogism
• Reference class forecasting