## 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.

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## 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.
- 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).

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## 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)*.

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

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### Keywords

- Imprecise probabilities
- Full conditional probabilities
- Independence