Abstract
We present and analyse four approaches to the reference class problem. First, we present a new objective Bayesian solution to the reference class problem. Second, we review Pollock’s combinatorial approach to the reference class problem. Third, we discuss a machine learning approach that is based on considering reference classes of individuals that are similar to the individual of interest. Fourth, we show how evidence of mechanisms, when combined with the objective Bayesian approach, can help to solve the reference class problem. We argue that this last approach is the most promising, and we note some positive aspects of the similarity approach.
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Notes
- 1.
The entropy of a probability function P defined on a set of logically independent propositions {E 1, …, E n } is defined by \(-\sum \limits _{i=1}^{n}P(E_{i})\log P(E_{i})\).
- 2.
- 3.
Let pe be a point. It is called peaking point with probability 1 iff for all δ > 0, PROB( | P ∗(A | S) − pe | < δ) = 1, i.e., for all ε > 0, PROB( | P ∗(A | S) − pe | < δ) > 1 −ε. If we think of δ being very small, for instance, 0. 000001, then this means that almost all subsets S are such that pe − 0. 000001 ≤ P ∗(A | S) ≤ pe + 0. 000001. Note that probability 1 does not mean that there are no exceptions, i.e., even if PROB( | P ∗(A | S) − pe | < δ) = 1, there are S such that | P ∗(A | S) − pe | > δ. However, such S are comparably few.
- 4.
For a related but more sophisticated similarity measure see Davis et al. (2010).
References
Benfante, R., K. Yano, L.-J. Hwang, J.D. Curb, A. Kagan, and W. Ross. 1994. Elevated serum cholesterol is a risk factor for both coronary heart disease and thromboembolic stroke in hawaiian Japanese men. Implications of shared risk. Stroke 25(4): 814–820.
Clarke, B., D. Gillies, P. Illari, F. Russo, and J. Williamson. 2014. Mechanisms and the evidence hierarchy. Topoi 33(2): 339–360.
Davis, D.A., N.V. Chawla, N.A. Christakis, and A.-L. Barabási. 2010. Time to care: A collaborative engine for practical disease prediction. Data Mining and Knowledge Discovery 20(3): 388–415.
Gower, J.C. 1971. A general coefficient of similarity and some of its properties. Biometrics 27(4): 857–871.
Haybittle, J., R. Blamey, C. Elston, J. Johnson, P. Doyle, F. Campbell, R. Nicholson, and K. Griffiths. 1982. A prognostic index in primary breast cancer. British Journal of Cancer 45(3): 361.
Hosseinpoor, A.R., L.A. Parker, E.T. d’Espaignet, and S. Chatterji. 2011. Social determinants of smoking in low-and middle-income countries: Results from the world health survey. PLoS One 6(5): e20331.
Illari, P.M., and J. Williamson. 2012. What is a mechanism? Thinking about mechanisms across the sciences. European Journal for Philosophy of Science 2: 119–135.
Jaynes, E.T. 1957. Information theory and statistical mechanics. The Physical Review 106(4): 620–630.
Jerez, J.M., I. Molina, P.J. García-Laencina, E. Alba, N. Ribelles, M. Martín, and L. Franco. 2010. Missing data imputation using statistical and machine learning methods in a real breast cancer problem. Artificial intelligence in medicine 50(2): 105–115.
Landes, J., and J. Williamson. (2016). Objective Bayesian nets from consistent datasets. In Proceedings of the 35th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, ed. A. Giffin, and K.H. Knuth, vol. 1757. American Institute of Physics Conference Proceedings, Potsdam
Mozaffarian, D., E.J. Benjamin, A.S. Go, D.K. Arnett, M.J. Blaha, M. Cushman, S. de Ferranti, J.-P. Despres, H.J. Fullerton, V.J. Howard, et al. 2015. Heart disease and stroke statistics-2015 update. A report from the American Heart Association. Circulation 131(4): E29–E322.
Parkkinen, V.-P., and J. Williamson. 2017. Extrapolating from model organisms in pharmacology. In Uncertainty in pharmacology: Epistemology, methods, and decisions, ed. B. Osimani. Dordrecht: Springer.
Pearl, J. 1988. Probabilistic reasoning in intelligent systems: Networks of plausible inference. San Mateo: Morgan Kaufmann.
Pollock, J.L. 2011. Reasoning defeasibly about probabilities. Synthese 181(2): 317–352.
Pollock, J.L. 2016. LISP code for OSCAR. http://johnpollock.us/ftp/OSCAR-web-page/oscar.html. Accessed 19 Sept 2016.
Russo, F., and J. Williamson 2007. Interpreting causality in the health sciences. International Studies in the Philosophy of Science 21(2): 157–170.
Shinton, R., and G. Beevers. 1989. Meta-analysis of relation between cigarette smoking and stroke. BMJ 298(6676): 789–794.
Statistik, Austria. 2016. Smoking statistics in Austria 2014. http://www.statistik.at/web_de/statistiken/menschen_und_gesellschaft/gesundheit/gesundheitsdeterminanten/rauchen/index.html. Accessed 28 Sept 2016.
Steel, D. 2008. Across the boundaries: Extrapolation in biology and social science. Oxford/New York: Oxford University Press.
Venn, J. 1888. The logic of chance, 3rd ed. London: Macmillan.
Wallmann, C. 2017. A Bayesian solution to the conflict of narrowness and precision in direct inference. Journal for General Philosophy of Science.
Wallmann, C., and G.D. Kleiter 2014a. Degradation in probability logic: When more information leads to less precise conclusions. Kybernetika 50(2): 268–283.
Wallmann, C., and G.D. Kleiter 2014b. Probability propagation in generalized inference forms. Studia Logica 102(4): 913–929.
Wang, X., Y. Dong, X. Qi, C. Huang, and L. Hou. 2013. Cholesterol levels and risk of hemorrhagic stroke. A systematic review and meta-analysis. Stroke 44(7): 1833–1839.
Wannamethee, S.G., A.G. Shaper, and S. Ebrahim. 2000. HDL-cholesterol, total cholesterol, and the risk of stroke in middle-aged British men. Stroke 31(8): 1882–1888.
Williamson, J. 2005a. Bayesian nets and causality: Philosophical and computational foundations. Oxford: Oxford University Press.
Williamson, J. 2005b. Objective Bayesian nets. In We will show them! essays in Honour of Dov Gabbay, ed. S. Artemov, H. Barringer, A.S. d’Avila Garcez, L.C. Lamb, and J. Woods, vol. 2, 713–730. London: College Publications.
Williamson, J. 2010. In defence of objective Bayesianism. Oxford: Oxford University Press.
Williamson, J. 2013. Why frequentists and Bayesians need each other. Erkenntnis 78(2): 293–318.
Williamson, J. 2017a. Lectures on inductive logic. Oxford: Oxford University Press.
Williamson, J. 2017b. Models in systems medicine. Disputatio. In press.
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The research for this paper has been funded by the Arts and Humanities Research Council via grant AH/M005917/1.
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Wallmann, C., Williamson, J. (2017). Four Approaches to the Reference Class Problem. In: Hofer-Szabó, G., Wroński, L. (eds) Making it Formally Explicit. European Studies in Philosophy of Science, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-55486-0_4
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