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, we could refer to the frequency of crashes among various different sets of aircraft: all aircraft, this make of aircraft, aircraft flown by this company in the last ten years, etc. In this example, the aircraft for which we wish to calculate the probability of a crash is a member of many different classes, in which the frequency of crashes differs. It is not obvious which class we should refer to for this aircraft. In general, any case is a member of very many classes among which the frequency of the attribute of interest differs. The reference class problem discusses which class is the most appropriate to use.
More formally, many arguments in statistics take the form of a statistical syllogism:
- proportion of are
- is an
- Therefore, the chance that is a is
is called the "reference class" and is the "attribute class" and is the individual object. How is one to choose an appropriate class ?
In Bayesian statistics, the problem arises as that of deciding on a prior probability for the outcome in question (or when considering multiple outcomes, a prior probability distribution).
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10. 11th Class Physics Lecture| Relative Motion | Rain Fall Cases | by Ashish Arora
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PROJECTION OF PLANE
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Instrument Flying Techniques 1961 US Air Force Pilot Training Film
Transcription
Lets analyze the observation of rain fall . by man on ground. Say if a man holding an umbrella and rain is falling vertically. Certainly this umbrella will be able to save man from the rain fall. But what happened if the man starts walking with the velocity v m. Say man starts walking we can easily calculate how the man will feel , the rain fall is coming . so if we calculate the velocity of, rain drops. With respect to man. Is velocity of rain drops with respect to man can be given as v r minus , v m. Now if we just calculate the direction of , this rain fall with respect to man , so we can simply state rain is falling in vertically downward direction say this v r vector, man is moving toward right with the velocity v m vector so minus v m vector would be this. So resultant will be in this direction it is a velocity of rain drops with respect to, man. That means as man will observe the rain fall man will fell that rain is falling at an angle theta from the vertical . where tan theta can be given as , the ratio of magnitude of v m and v r. This the angle at which the rain appears to be coming, with respect to man. So if while walking he is holding the umbrella in vertical direction , and rain drops are hitting to the man in the direction theta with the vertical , certainly the man is going to be wet. So to save, himself from the rainfall he has to hold the umbrella , in such a way , that it’ll be able to , stop the rain drops with respect to man hitting to him . like say if this is the man , so he has to hold the umbrella , in this direction , where we can say , rain drops, appear to be coming towards the man . this is a velocity of rain with respect to man , at an angle theta with the vertical with which the rain appear to be falling . so even, there is no umbrella over head, still he’ll not get wet because , the rain drops which is coming, towards him , it’ll be saved by the umbrella , and it’ll be going away. The rain drops which is falling on to it when it’ll fall before that he’ll be able to move forward , and umbrella will be able to save the rain drops, which are going to hit the man in vertical direction . so this the way how he has to hold the umbrella to save himself , actually umbrella will be able to save the man if it is in direction, which is opposite to the velocity of rain drops , with respect to ,man. In this example we are given that a man is standing on ground with an umbrella hat as shown. Suddenly rain fall starts at a speed 20 meters per second at an angle 30 degree with the vertical . we are required to find the speed and direction in which man should start running so as to save himself. In this situation, we can see that, the man is wearing an umbrella hat. This umbrella hat is very small in size and it can save man only when rain is, falling vertically . when rain fall starts in vertical direction . but here rain is falling in the direction at an angle theta with the vertical , so obviously this umbrella hat will not be able to save him. In this case the man should start running, in such a way that velocity of rain, with respect to this man should become vertically down , than this umbrella will be able to save him. Say here you can simply write down in the solution , we can write , to save. The man. He should run. Such that. Velocity of rain with respect to man , becomes vertically down. If it becomes vertically down, this umbrella will be able to, save him. So here we know velocity of rain with respect to man we can write as v r minus, v m. We are required to find the value of v m. So here v m can be given as, v r minus v r m. So in this situation we can see that v r in this direction , this the velocity v r , at an angle, theta i.e.30 degree with the vertical and the value of v r is also given. If v r m is in vertically downward direction , if rain with respect to man is in vertically downward direction, then minus v r m will be in vertically upward direction , so the resultant will be toward left, which is velocity of man. So man has to run toward left with the velocity v m , and this v m, magnitude can be directly given by this triangle, as v m =, v r sine theta , i.e. = 20 sine 30 degree or it is 10 meters per second . this is the answer to this problem . so man has to run toward left, with the velocity 10 meters per second , due to this, it’ll appear to man that rain is falling vertically , so this umbrella hat will be able to save the man . or we can say or we can solve this problem, orally or directly also. like if rain is falling at a velocity v r. Its vertical component can be written as v r coz theta and its horizontal component with which the rain is, travelling , or rain drops are travelling in horizontal direction as v r sine theta. So, if we talk about vertical component of rain drops umbrella is able to save the man. For horizontal component man will not be saved , so he has to run in such a way that horizontal speed of rain, appear to man will be= 0. So we can directly say the man has to run toward left with the velocity v r sine theta, so that the relative velocity of rain drops with respect to man in horizontal , direction become 0. This the way how you can directly archive the result, or this is the systematic way to approach the result .
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 philosophy[3] and in the life sciences, e.g., clinical trial prediction.[4]
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.[5][6]
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.[7][8]
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.
- ^ 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". Rochester, NY. SSRN 3981732.
{{cite journal}}: Cite journal requires|journal=(help) - ^ 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
- ^ 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.
- ^ Franklin, James (Mar 2010). "Feature selection methods for solving the reference class problem" (PDF). Columbia Law Review Sidebar. 110. Retrieved 30 June 2021.
- ^ Franklin, James (2011). "The objective Bayesian conceptualisation of proof and reference class problems". Sydney Law Review. 33: 545–561. Retrieved 30 June 2021.
This page was last edited on 4 March 2024, at 18:03