Abstract
This chapter explores the topic of imprecise probabilities (IP) as it relates to model validation. IP is a family of formal methods that aim to provide a better representation of severe uncertainty than is possible with standard probabilistic methods. Among the methods discussed here are using sets of probabilities to represent uncertainty, and using functions that do not satisfy the additvity property. We discuss the basics of IP, some examples of IP in computer simulation contexts, possible interpretations of the IP framework and some conceptual problems for the approach. We conclude with a discussion of IP in the context of model validation.
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Notes
- 1.
Although one can find precedents going back to Keynes or even Boole, IP really started in the middle of the twentieth century with work by people like Koopman (1940), Good (1952, 1962), Smith (1961) and Dempster (1967). Work in philosophy on IP really starts with Levi (1974, 1980, 1986). Important formal and philosophical work on IP was carried out by Seidenfeld (1983, 1988); Seidenfeld et al. (1989) (Seidenfeld was Levi’s graduate student). Walley (1991) was a hugely influential book which, until recently, was still the go-to monograph for many formal details of the theory. The state of the art in terms of formal theory can be found in Augustin et al. (2014) and Troffaes and de Cooman (2014). Bradley (2014) provides a philosophical overview.
- 2.
Consider some \(\mathbf {pr}\in \mathscr {P}\) for which \(\mathbf {pr}(X\vee Y) = \underline{\mathscr {P}}(X\vee Y)\). \(\inf \mathscr {P}(X) + \inf \mathscr {P}(Y) \le \mathbf {pr}(X) + \mathbf {pr}(Y)\), since \(\mathbf {pr}\in \mathscr {P}\), so \(\underline{\mathscr {P}}\) is superadditive. Boundedness is trivial, and much the same reasoning works if the set \(\mathscr {P}\) doesn’t attain its bounds (just think in terms of the closure of the set).
- 3.
There is a one-to-one correspondence between lower probabilities and a subset of the set of credal sets, namely those with some nice topological properties. We don’t need to discuss this here, but see the above-listed references for details.
- 4.
See Oberkampf and Helton (2004) for a discussion of DS theory in an engineering context.
- 5.
For introductions to these aspects of IP, see Augustin et al. (2014) Chapters 7, 9 and 11, respectively.
- 6.
I am drawing mainly from Sect.13.4, but similar ideas appear in a number of other places in the book.
- 7.
- 8.
- 9.
For more on the interpretation of IP, see Bradley (2014).
- 10.
We earlier described probability theory in terms of events rather than random variables, but the difference is mostly cosmetic. Real-valued random variables are functions from events to real numbers, events are “indicator functions” in the space of random variables.
- 11.
For example, if you find f desirable, and you find \(f'\) desirable, you should find \(f+f'\) desirable; or if a gamble’s payout is always non-negative, then it is desirable.
- 12.
- 13.
See also Oberkampf and Helton (2004) for an example of DS theory in an engineering context.
- 14.
The restriction to non-zero probability in the conditioning event is for convenience: if we had defined credal sets in terms of Popper functions or similar we could do without such a restriction.
- 15.
A recent characterisation of dilation is found in Pedersen and Wheeler (2014).
- 16.
This description of the puzzle follows Joyce (2010).
- 17.
- 18.
See Bradley (2015) for some discussion of the options.
- 19.
See Hájek (2011) for an introduction to interpretations of probability.
- 20.
For a more careful and rigorous characterisation of dilation, see Pedersen and Wheeler (2014).
References
Augustin, T., Coolen, F. P., de Cooman, G., & Troffaes, M. C. (Eds.) (2014). Introduction to imprecise probabilities. John Wiley and Sons.
Bradley, S. (2014). Imprecise probabilities. In E. N. Zalta (ed.), The Stanford encyclopedia of philosophy.
Bradley, S. (2015). How to choose among choice functions. In ISIPTA 2015 proceedings (pp. 57–66).
Bradley, S. (2016). Vague chance. Ergo, 3(20)
Bradley, S., & Steele, K. (2014a). Should subjective probabilities be sharp? Episteme, 11, 277–289.
Bradley, S., & Steele, K. (2014b). Uncertainty, learning and the “problem” of dilation. Erkenntnis, 79, 1287–1303.
Chandler, J. (2014). Subjective probabilities need not be sharp. Erkenntnis, 79, 1273–1286.
Cozman, F. (2012). Sets of probability distributions, independence and convexity. Synthese, 186, 577–600.
de Cooman, G., & Troffaes, M. (2004). Coherent lower previsions in systems modelling: Products and aggregation rules. Reliability Engineering and System Safety, 85, 113–134.
Dempster, A. (1967). Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics, 38, 325–339.
Elga, A. (2010). Subjective probabilities should be sharp. Philosophers’ Imprint, 10.
Ferson, S., & Ginzburg, L. R. (1996). Different methods are needed to propagate ignorance and variability. Reliability Engineering and System Safety, 54, 133–144.
Ferson, S., & Hajagos, J. G. (2004). Arithmetic with uncertain numbers: Rigorous and (often) best possible answers. Reliability Engineering and System Safety, 85, 135–152.
Ferson, S., Joslyn, C. A., Helton, J. C., Oberkampf, W. L., & Sentz, K. (2004). Summary from the epistemic uncertainty workshop: Consensus amid diversity. Reliability Engineering and System Safety, 85, 355–369.
Fetz, T., & Oberguggenberger, M. (2004). Propagation of uncertainty through multivariate functions in the framework of sets of probability measures. Reliability Engineering and System Safety, 85, 73–87.
Frigg, R., Bradley, S., Du, H., & Smith, L. A. (2014). Laplace’s demon and the adventures of his apprentices. Philosophy of Science, 81, 31–59.
Gong, R., & Meng, X. -L. (2017). Judicious judgment meets unsettling update: Dilation, sure loss and simpson’s paradox. arXiv:1712.08946v1.
Good, I. J. (1952). Rational decisions. Journal of the Royal Statistical Society Series B (Methodological), 14, 107–114.
Good, I. J. (1962). Subjective probability as the measure of a non-measurable set. In Logic, methodology and philosophy of science: Proceedings of the 1960 international congress (pp. 319–329).
Hájek, A. (2011). Interpretations of probability. In E. N. Zalta (ed.), The stanford encyclopedia of philosophy. Stanford. http://plato.stanford.edu/archives/fall2007/entries/probability-interpret/.
Halpern, J. Y. (2003). Reasoning about uncertainty. MIT Press.
Hart, C., & Titelbaum, M. G. (2015). Intuitive dilation? Thought, 4, 252–262.
Hawthorne, J. (2005). Degree-of-belief and degree-of-support: Why Bayesians need both notions. Mind, 114, 277–320.
Helton, J. C., Johnson, J., & Oberkampf, W. L. (2004). An exploration of alternative approaches to the representation of uncertainty in model predictions. Reliability Engineering and System Safety, 85, 39–71.
Helton, J. C., & Oberkampf, W. L. (2004). Alternative representations of epistemic uncertainty. Reliability Engineering and System Safety, 85, 1–10.
Joyce, J. M. (2010). A defense of imprecise credence in inference and decision. Philosophical Perspectives, 24, 281–323.
Klir, G. J., & Smith, R. M. (2001). On measuring uncertainty and uncertainty-based information: Recent developments. Annals of Mathematics and Artificial Intelligence, 32, 5–33.
Koopman, B. O. (1940). The bases of probability. Bulletin of the American Mathematical Society, 46, 763–774.
Levi, I. (1974). On indeterminate probabilities. Journal of Philosophy, 71, 391–418.
Levi, I. (1980). The enterprise of knowledge. The MIT Press.
Levi, I. (1986). Hard Choices: Decision making under unresolved conflict. Cambridge University Press.
Morgan, M. G. & Henrion, M. (1990). Uncertainty: A guide to dealing with uncertainty in quantitiative risk and policy analysis. Cambridge University Press.
Oberkampf, W. L., & Helton, J. C. (2004). Evidence theory for engineering applications. In Engineering design reliability handbook (pp. 197–226). CRC Press.
Oberkampf, W. L., Helton, J. C., Joslyn, C. A., Wojtkiewicz, S. F., & Ferson, S. (2004). Challenge problems: Uncertainty in system response given uncertain parameters. Reliability Engineering and System Safety, 85, 11–19.
Oberkampf, W. L., & Roy, C. J. (2010). Verification and validation in scientific computing. Cambridge University Press.
Pedersen, A. P., & Wheeler, G. (2014). Demystifying dilation. Erkenntnis, 79, 1305–1342.
Rinard, S. (2013). Against radical credal imprecision. Thought, 2, 157–165.
Rinard, S. (2015). A decision theory for imprecise probabilities. Philosophers’ Imprint, 15, 1–16.
Ruggeri, F., Insua, D. R., & Martín, J. (2005). Robust bayesian analysis. In Handbook of statistics (Vol. 25, pp. 623–667). Elsevier.
Sahlin, N.-E., & Weirich, P. (2014). Unsharp sharpness. Theoria, 80, 100–103.
Seidenfeld, T. (1983). Decisions with indeterminate probabilities. The Brain and Behavioural Sciences, 6, 259–261.
Seidenfeld, T. (1988). Decision theory without “independence” or without “ordering”. what’s the difference? Economics and philosophy (pp. 267–290).
Seidenfeld, T., Kadane, J. B., & Schervish, M. J. (1989). On the shared preferences of two Bayesian decision makers. The Journal of Philosophy, 86, 225–244.
Seidenfeld, T., & Wasserman, L. (1993). Dilation for sets of probabilities. Annals of Statistics, 21, 1139–1154.
Smith, C. A. (1961). Consistency in statistical inference and decision. Journal of the Royal Statistical Society Series B (Methodological), 23, 1–37.
Smith, V. L. & van Boening, M. (2008). Exogenous uncertainty increases the bid-ask spread in the continuous double auction. In Handbook of experimental economics results (vol. 1). Elsevier.
Stainforth, D. A., Allen, M. R., Tredger, E., & Smith, L. A. (2007a). Confidence uncertainty and decision-support relevance in climate models. Philosophical Transactions of the Royal Society, 365, 2145–2161.
Stainforth, D. A., Downing, T., Washington, R., Lopez, A., & New, M. (2007b). Issues in the interpretation of climate model ensembles to inform decisions. Philosophical Transactions of the Royal Society, 365, 2163–2177.
Troffaes, M., & de Cooman, G. (2014). Lower previsions. Wiley.
Vallinder, A. (2018). Imprecise bayesianism and global belief inertia. The British Journal for the Philosophy of Science, 69, 1205–1230.
Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities, volume 42 of Monographs on Statistics and Applied Probability. Chapman and Hall.
White, R. (2010). Evidential symmetry and mushy credence. In T. Szabo Gendler, & J. Hawthorne (Eds.), Oxford studies in epistemology (pp. 161–186). Oxford University Press.
Williamson, T. (1994). Vagueness. Routledge.
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Bradley, S. (2019). Imprecise Probabilities. In: Beisbart, C., Saam, N. (eds) Computer Simulation Validation. Simulation Foundations, Methods and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-70766-2_21
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