Abstract
Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. This is a powerful generalisation of de Finetti’s representation theorem allowing for both imprecision and indecision. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Many inference principles encountered in the literature can then be interpreted, and represented mathematically, as restrictions on such maps. We discuss two important inference principles: representation insensitivity—a strengthened version of Walley’s representation invariance—and specificity. We show that there is a infinity of inference systems that satisfy these two principles, amongst which we discuss in particular the inference systems corresponding to (a modified version of) Walley and Bernard’s imprecise Dirichlet multinomial models (IDMMs) and the Haldane inference system.
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Notes
- 1.
… unless the observed sequence has probability zero.
- 2.
To avoid confusion, we make a (perhaps non-standard) distinction between the multinomial expectation, which is associated with sequences of observations, and the count multinomial expectation, associated with their count vectors.
- 3.
The degree may be smaller than n because the sum of all Bernstein basis polynomials of fixed degree is one. Strictly speaking, these polynomials p are restrictions to \(\Sigma_{A}\) of multivariate polynomials q on \(\mathbb{R}^A\), called representations of p. For any p, there are multiple representations, with possibly different degrees. The smallest such degree is then called the degree deg\((p)\) of p.
- 4.
Strictly speaking, Eq. 2.7 only defines the count multinomial expectation operator \(\text{CoMn}^{n}_{A}\) for \(n>0\), but it is clear that the definition extends trivially to the case n = 0.
- 5.
Actually, a suitably adapted version of coherence, where the gambles are restricted to the polynomials on \(\Sigma_{A}\).
- 6.
It is an immediate consequence of the F. Riesz extension theorem that each such linear prevision is the restriction to polynomials of the expectation operator of some unique σ-additive probability measure on the Borel sets of \(\Sigma_{A}\); see for instance [6].
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Acknowledgements
Gert de Cooman’s research was partially funded through project number 3G012512 of the Research Foundation Flanders (FWO). Jasper De Bock is a PhD Fellow of the Research Foundation Flanders and wishes to acknowledge its financial support. Marcio Diniz was supported by FAPESP (São Paulo Research Foundation), under the project 2012/14764-0 and wishes to thank the SYSTeMS Research Group at Ghent University for its hospitality and support during his sabbatical visit there.
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de Cooman, G., De Bock, J., Diniz, M. (2015). Predictive Inference Under Exchangeability, and the Imprecise Dirichlet Multinomial Model. In: Polpo, A., Louzada, F., Rifo, L., Stern, J., Lauretto, M. (eds) Interdisciplinary Bayesian Statistics. Springer Proceedings in Mathematics & Statistics, vol 118. Springer, Cham. https://doi.org/10.1007/978-3-319-12454-4_2
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