Reference class problem

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.

Legal applications
Applying Bayesian probability in practice involves assessing a prior probability which is then applied to a likelihood function and updated through the use of Bayes' theorem. Suppose we wish to assess the probability of guilt of a defendant in a court case in which DNA (or other probabilistic) evidence is available. We first need to assess the prior probability of guilt of the defendant. We could say that the crime occurred in a city of 1,000,000 people, of whom 15% meet the requirements of being the same sex, age group and approximate description as the perpetrator. That suggests a prior probability of guilt of 1 in 150,000. We could cast the net wider and say that there is, say, a 25% chance that the perpetrator is from out of town, but still from this country, and construct a different prior estimate. We could say that the perpetrator could come from anywhere in the world, and so on.

Legal theorists have discussed the reference class problem particularly with reference to the Shonubi case. Charles Shonubi, a Nigerian drug smuggler, was arrested at JFK Airport on Dec 10, 1991, and convicted of heroin importation. The severity of his sentence depended not only on the amount of drugs on that trip, but the total amount of drugs he was estimated to have imported on seven previous occasions on which he was not caught. Five separate legal cases debated how that amount should be estimated. In one case, "Shonubi III", the prosecution presented statistical evidence of the amount of drugs found on Nigerian drug smugglers caught at JFK Airport in the period between Shonubi's first and last trips. There has been debate over whether that is the (or a) correct reference class to use, and if so, why.

Other legal applications involve valuation. For example, houses might be valued using the data in a database of house sales of "similar" houses. To decide on which houses are similar to a given one, one needs to know which features of a house are relevant to price. Number of bathrooms might be relevant, but not the eye color of the owner. It has been argued that such reference class problems can be solved by finding which features are relevant: a feature is relevant to house price if house price covaries with it (it affects the likelihood that the house has a higher or lower value), and the ideal reference class for an individual is the set of all instances which share with it all relevant features.