Skip to main content
Log in

On the imprecision of full conditional probabilities

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

The purpose of this paper is to show that if one adopts conditional probabilities as the primitive concept of probability, one must deal with the fact that even in very ordinary circumstances at least some probability values may be imprecise, and that some probability questions may fail to have numerically precise answers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Jeffrey Sanford Russell, John Hawthorne & Lara Buchak

Notes

  1. In particular, see the axiomatizations of Renyi (1955) and Popper (1959), and Dubins’s extension of de Finetti (1975). A sample of authors endorsing full conditional probabilities includes representatives from philosophy (van Fraassen 1976; Levi 1980; McGee 1994; Hájek 2003; Sprenger and Hartmann 2019), statistics (Kadane et al. 1999), economics and game theory (Blume et al. 1991b; Myerson 1991; Hammond 1994; Battigalli and Veronesi 1996; Kohlberg and Reny 1997), logic (Adams 1966; Coletti and Scozzafava 2002; Makinson 2011), psychology (Pfeifer and Tulkki 2017), and computer science (Kraus et al. 1990; Cowell et al. 1999; Gilio 2012).

  2. A sample of recent objections to imprecise probabilities include White (2010), Elga (2010) and Mahtani (2017). For a survey of responses see Walley (1991, §§5.6–5.9) and also Joyce (2011), Pedersen and Wheeler (2014) and Chandler (2014).

  3. This type of difficulty has been noted earlier by Kohlberg and Reny in connection with their definition of strong independence (Kohlberg and Reny 1997, Remark 7) and by Cozman with respect to the specification of Bayesian networks (Cozman 2013). Our aim here is to show that imprecision in probability values is a fundamental feature of full conditional probability, not a narrow technical issue that appears in some accounts of full conditional probability but not others, nor a property that appears only under some notions of independence but not others.

  4. Briefly, a Bayesian network is a pair consisting of a directed acyclic graph and a probability distribution (Pearl 1988). The graph consists of nodes and edges, and each node is a random variable. The graph and the distribution are related by the following Markov condition: a random variable V is independent of its nondescendants given its parents (a parent of V is a node U such that there is an edge from U to V; a descendant of V is a node U such that there is a directed path from U to V). Consequently, the joint distribution over all random variables factorizes into local conditional distributions: each node/variable V is associated with the probability values \({\mathbb {P}}\!\left( V=v|\mathrm {pa}(V)=\pi \right) \), for each value v of V and each value \(\pi \) of \(\mathrm {pa}(V)\), the parents of V.

  5. As Giron and Rios do, for example, as their preferences are undefined when conditioned on events of zero probabilities (Giron and Rios 1980).

References

  • Adams, E. W. (1966). Probability and the logic of conditionals. In J. Hintikka & P. Suppes (Eds.), Aspects of inductive logic. Amsterdam: North Holland.

    Google Scholar 

  • Battigalli, P., & Veronesi, P. (1996). A note on stochastic independence without Savage-null events. Journal of Economic Theory, 70(1), 235–248.

    Article  Google Scholar 

  • Blume, L., Brandenburger, A., & Dekel, E. (1991a). Lexicographic probabilities and choice under uncertainty. Econometrica, 58(1), 61–79.

    Article  Google Scholar 

  • Blume, L., Brandenburger, A., & Dekel, E. (1991b). Lexicographic probabilities and equilibrium refinements. Econometrica, 58(1), 81–98.

    Article  Google Scholar 

  • Chandler, J. (2014). Subjective probabilities need not be sharp. Erkenntnis, 7(6), 1273–1286.

    Article  Google Scholar 

  • Coletti, G., & Scozzafava, R. (2002). Probabilistic logic in a coherent setting, trends in logic (Vol. 15). Dordrecht: Kluwer.

    Book  Google Scholar 

  • Cowell, R. G., Dawid, A. P., Lauritzen, S. L., & Spiegelhalter, D. J. (1999). Probabilistic networks and expert systems. New York: Springer.

    Google Scholar 

  • Cozman, F. G. (2013). Independence for full conditional probabilities: Structure, factorization, non-uniqueness, and Bayesian networks. International Journal of Approximate Reasoning, 54, 1261–1278.

    Article  Google Scholar 

  • Cozman, F. G., & Seidenfeld, T. (2009). Independence for full conditional measures and their graphoid properties. In B. Lowe, E. Pacuit, & J.-W. Romeijn (Eds.), Reasoning about probabilities and probabilistic reasoning, volume 16 of foundations of the formal sciences VI (pp. 1–29). London: College Publications.

    Google Scholar 

  • de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré, 7(1), 1–68. Trans. Henry E. Kyburg, Jr., Foresight: Its logical laws, its subjective sources. In H. E. Kyburg Jr. & H. Smokler (Eds.), Studies in subjective probability. New York: Wiley (1964).

  • de Finetti, B. (1949). Sull’impostazione assiomatica del calcolo delle probabilità. Annali Triestini dell’Università 19, 19–29. Trans. G. Machi and A. Smith, On the axiomatization of probability theory. In B. de Finetti, Probability, induction and statistics: The art of guessing. London: Wiley (1972).

  • de Finetti, B. (1974). Theory of probability (Vol. 1–2). New York: Wiley.

  • de Finetti, B., & Savage, L. J. (1962). Sul modo di scegliere le probabilità iniziali. Biblioteca del Metron, Serie C, 1, 81–154.

    Google Scholar 

  • Dizadji-Bahmani, F., Frigg, R., & Hartmann, S. (2011). Confirmation and reduction: A Bayesian account. Synthese, 179, 321–338.

    Article  Google Scholar 

  • Dubins, L. E. (1975). Finitely additive conditional probability, conglomerability and disintegrations. Annals of Statistics, 3(1), 89–99.

    Google Scholar 

  • Easwaran, K. (2011). Varieties of conditional probability. In P. Bandyopadhyay & M. Forster (Eds.), Philosophy of statistics, volume 7 of handbook of the philosophy of science. Amsterdam: North Holland.

    Google Scholar 

  • Elga, A. (2010). Subjective probabilities should be sharp. Philosophers Imprint, 10(5), 1–11.

    Google Scholar 

  • Gilio, A. (2012). Generalizing inference rules in a coherence-based probabilistic default reasoning. International Journal of Approximate Reasoning, 53(3), 413–434.

    Article  Google Scholar 

  • Giron, F. J., & Rios, S. (1980). Quasi-Bayesian behaviour: A more realistic approach to decision making? In J. M. Bernardo, J. H. DeGroot, D. V. Lindley, & A. F. M. Smith (Eds.), Bayesian statistics (pp. 17–38). Valencia: University Press.

    Google Scholar 

  • Hájek, A. (2003). What conditional probability could not be. Synthese, 137, 273–323.

    Article  Google Scholar 

  • Hammond, P. J. (1994). Elementary non-Archimedean representations of probability for decision theory and games. In P. Humphreys (Ed.), Patrick Suppes: Scientific philosopher (Vol. 1, pp. 25–59). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Joyce, J. (2011). A defense of imprecise credences in inference and decision making. Philosophical Perspectives, 24(1), 281–323.

    Article  Google Scholar 

  • Kadane, J. B., Schervish, M. J., & Seidenfeld, T. (1999). Rethinking the foundations of statistics, Cambridge series in probability, induction and decision theory. Cambridge: Cambridge University Press.

    Google Scholar 

  • Keynes, J. M. (1921). A treatise on probability. London: Macmillan and Co.

    Google Scholar 

  • Knight, F. H. (1921). Risk, uncertainty, and profit. Boston: Hart, Schaffner & Marx; Houghton Mifflin Company.

    Google Scholar 

  • Kohlberg, E., & Reny, P. J. (1997). Independence on relative probability spaces and consistent assessments in game trees. Journal of Economic Theory, 75, 280–313.

    Article  Google Scholar 

  • Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 14(1), 167–207.

    Article  Google Scholar 

  • Levi, I. (1974). On indeterminate probabilities. Journal of Philosophy, 71, 391–418.

    Article  Google Scholar 

  • Levi, I. (1980). The enterprise of knowledge. Cambridge, MA: MIT Press.

    Google Scholar 

  • Levi, I. (2002). Indeterminate probability and change of view. In S. Haller and G. Simmons (Eds.), Proceedings of the 15th international florida artificial intelligence research society conference (FLAIRS 2002) (pp. 503–507). AAAI.

  • Mahtani, A. (2017). Imprecise probabilities and unstable betting behavior. Noûs, 52, 69–87.

    Article  Google Scholar 

  • Makinson, D. (2011). Conditional probability in the light of qualitative belief change. Journal of Philosophical Logic, 40(2), 121–153.

    Article  Google Scholar 

  • McGee, V. (1994). Learning the impossible. In E. Bells & B. Skyrms (Eds.), Probability and conditionals (pp. 179–199). Cambridge: Cambridge University Press.

    Google Scholar 

  • Myerson, R. B. (1991). Game theory: Analysis of conflict. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Pearl, J. (1988). Probabilistic reasoning in intelligent systems: Networks of plausible inference. San Mateo, CA: Morgan Kaufmann.

    Google Scholar 

  • Pedersen, A. P., & Wheeler, G. (2014). Demystifying dilation. Erkenntnis, 79(6), 1305–1342.

    Article  Google Scholar 

  • Pedersen, A. P., & Wheeler, G. (2015). Dilation, disintegrations, and delayed decisions. In Proceedings of the 9th symposium on imprecise probabilities and their applications (ISIPTA), Pescara, Italy (pp. 227–236).

  • Pedersen, A. P., & Wheeler, G. (2019). Dilation and asymmetric relevance. In J. De Bock, C. P. Campos, G. de Cooman, E. Quaeghebeur, & G. Wheeler (Eds.), Proceedings of machine learning research, volume 103 of proceedings of the 11th symposium on imprecise probabilities and their applications (ISIPTA) (pp. 324–326).

  • Pfeifer, N., & Tulkki, L. (2017). Conditionals, counterfactuals, and rational reasoning: An experimental study on basic principles. Minds and Machines, 1, 119–165.

    Article  Google Scholar 

  • Pollard, D. (2002). A user’s guide to measure theoretic probability. Cambridge: Cambridge University Press.

    Google Scholar 

  • Popper, K. R. (1959). The logic of scientific discovery. London: Routledge.

    Google Scholar 

  • Quaeghebeur, E. (2014). Desirability. In T. Augustin, F. P. A. Coolen, G. de Cooman, & M. C. M. Troffaes (Eds.), Introduction to imprecise probabilities (pp. 1–27). New York: Wiley.

    Google Scholar 

  • Renyi, A. (1955). On a new axiomatic theory of probability. Acta Mathematica Academiae Scientiarum Hungarian, 6, 285–335.

    Article  Google Scholar 

  • Seidenfeld, T., Schervish, M. J., & Kadane, J. B. (2001). Improper regular conditional distributions. The Annals of Probability, 29(4), 1612–1624.

    Google Scholar 

  • Seidenfeld, T., & Wasserman, L. (1993). Dilation for sets of probabilities. Annals of Statistics, 21(9), 1139–1154.

    Google Scholar 

  • Spohn, W. (1986). The representation of Popper measures. Topoi, 5(1), 69–74.

    Article  Google Scholar 

  • Sprenger, J., & Hartmann, S. (2019). Bayesian philosophy of science. Oxford: Oxford University Press.

  • Swinkels, J. M. (1993). Independence for conditional probability systems. Technical Report 1076, Northwestern University, Center for Mathematical Studies in Economics and Management Science.

  • van Fraassen, B. C. (1976). Representation of conditional probabilities. Journal of Philosophical Logic, 5, 417–430.

    Article  Google Scholar 

  • Vantaggi, B. (2001). Conditional independence in a coherent finite setting. Annals of Mathematics and Artificial Intelligence, 32(1–4), 287–313.

    Article  Google Scholar 

  • Vicig, P., & Seidenfeld, T. (2012). Bruno de Finetti and imprecision: Imprecise probability does not exist!. International Journal of Approximate Reasoning, 53, 1115–1123.

    Article  Google Scholar 

  • Walley, P. (1991). Statistical reasoning with imprecise probabilities. London: Chapman and Hall.

    Book  Google Scholar 

  • Walley, P. (2000). Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning, 24, 125–148.

    Article  Google Scholar 

  • Weichselberger, K. (2000). The theory of interval-probability as a unifying concept for uncertainty. International Journal of Approximate Reasoning, 24(2–3), 149–170.

    Article  Google Scholar 

  • Wheeler, G. (2021). A gentle approach to imprecise probability. In T. Augustin, F. Cozman, & G. Wheeler (Eds.), Essays in Honor of Teddy Seidenfeld, Theory and decision library A. Dordrecht: Springer.

    Google Scholar 

  • White, R. (2010). Evidential symmetry and mushy credence. In T. S. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 3, pp. 161–186). Oxford: Oxford University Press.

    Google Scholar 

  • Williams, P. M. (1975). Notes on conditional previsions. School of Mathematical and Physical Sciences, University of Sussex. Republished in International Journal of Approximate Reasoning, 44(3): 366–383, 2007.

Download references

Acknowledgements

Gregory Wheeler’s research was supported in part by the joint Agence Nationale de la Recherche (ANR) & Deutsche Forschungsgemeinschaft (DFG) project “Collective Attitudes Formation” ColAForm, award RO 4548/8-1, and Fabio Cozman’s by awards from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Funda de Amparo squisa do São Paulo (FAPESP).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gregory Wheeler.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wheeler, G., Cozman, F.G. On the imprecision of full conditional probabilities. Synthese 199, 3761–3782 (2021). https://doi.org/10.1007/s11229-020-02954-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-020-02954-z

Keywords

Navigation