In statistics, the reference class problem involves choosing the appropriate class to calculate the probability relevant to a specific case. For example, estimating an aircraft’s crash probability depends on whether we consider all aircraft, a specific make, or those operated by a certain company. Each class yields different frequencies, making the choice unclear. This is framed by the statistical syllogism, where the "reference class" (F) and the "attribute class" (G) define probabilities for an individual (I). In Bayesian statistics, this problem translates to selecting an appropriate prior probability. Thus, deciding the proper reference class is crucial for accurate probability assessments.
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.1
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."2
There has also been discussion of the reference class problem in philosophy3 and in the life sciences, e.g., clinical trial prediction.4
In the book Anthropic Bias, philosopher Nick Bostrom described ways in which reference classes can be applied to reasoning about one's position in reality. Bostrom investigates how to reason when one suspects that evidence is biased by "observation selection effects", in other words, when the evidence presented has been pre-filtered by the condition that there was some appropriately positioned observer to "receive" the evidence.56 This conundrum is sometimes called the "anthropic principle", "self-locating belief", or "indexical information". The book first discusses the fine-tuned universe hypothesis and its possible explanations, notably considering the possibility of a multiverse.
Bostrom argues against the self-indication assumption (SIA), a term he uses to characterize some existing views, and introduces the self-sampling assumption (SSA): that you should think of yourself as if you were a random observer from a suitable reference class. He later refines SSA into using observer-moments instead of observers to address certain paradoxes in anthropic reasoning, formalized as the strong self-sampling assumption (SSSA): Each observer-moment should reason as if it were randomly selected from the class of all observer-moments in its reference class.7 These different assumptions are affected differently based on the choice of reference class. An application of the principle underlying SSSA (though this application is nowhere expressly articulated by Bostrom), is: If the minute in which you read this article is randomly selected from every minute in every human's lifespan, then (with 95% confidence) this event has occurred after the first 5% of human observer-moments. If the mean lifespan in the future is twice the historic mean lifespan, this implies 95% confidence that N < 10n (the average future human will account for twice the observer-moments of the average historic human). Therefore, the 95th percentile extinction-time estimate in this version is 4560 years.
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.89
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.1011
See also
References
J. Venn,The Logic of Chance (2nd ed, 1876), p. 194. ↩
H. Reichenbach, The Theory of Probability (1949), p. 374 ↩
A. Hájek, The Reference Class Problem is Your Problem Too, Synthese 156 (2007): 185-215. https://web.archive.org/web/20180111165141/https://pdfs.semanticscholar.org/2899/90fa6af58b0e1b3103fdee05aef57e53de48.pdf ↩
Atanasov, Pavel D.; Joseph, Regina; Feijoo, Felipe; Marshall, Max; Siddiqui, Sauleh (2021-12-09). "Human Forest vs. Random Forest in Time-Sensitive COVID-19 Clinical Trial Prediction". SSRN Electronic Journal. Rochester, NY. SSRN 3981732. https://papers.ssrn.com/abstract=3981732 ↩
"Anthropic Bias | anthropic-principle.com". www.anthropic-principle.com. Retrieved 2015-11-03. http://www.anthropic-principle.com/?q=anthropic_bias ↩
Bostrom, Nick (2002). Anthropic bias: observation selection effects in science and philosophy. Studies in philosophy. New York: Routledge. ISBN 978-0-415-88394-8. 978-0-415-88394-8 ↩
Bostrom, Nick (2005). "Self-Location and Observation Selection Theory". anthropic-principle.com. Retrieved 2023-07-02. https://anthropic-principle.com/preprints/self-location ↩
M. Colyvan, H.M. Regan and S. Ferson, Is it a crime to belong to a reference class?, Journal of Political Philosophy 9 (2001): 168-181 http://colyvan.com/papers/shonubi.pdf ↩
Tillers, Peter (2005). "If wishes were horses: discursive comments on attempts to prevent individuals from being unfairly burdened by their reference classes". Law, Probability and Risk. 4 (1–2): 33–49. doi:10.1093/lpr/mgi001. /wiki/Peter_Tillers ↩
Franklin, James (Mar 2010). "Feature selection methods for solving the reference class problem" (PDF). Columbia Law Review Sidebar. 110. Retrieved 30 June 2021. /wiki/James_Franklin_(philosopher) ↩
Franklin, James (2011). "The objective Bayesian conceptualisation of proof and reference class problems". Sydney Law Review. 33: 545–561. Retrieved 30 June 2021. http://www8.austlii.edu.au/cgi-bin/viewdoc/au/journals/SydLawRw/2011/23.html ↩