**Books, papers in books **

G. de Cooman and E. Miranda, *Symmetry
of models versus models of symmetry. *In:
Probability and Inference: essays in Honour of Henry E. Kyburg (W. Harper and G.
Wheeler, eds.), pp. 67--149. College Publications, 2007.

Abstract: A model for a
subject beliefs about a certain phenomenon may exhibit symmetry, in the sense
that it is invariant under certain transformations. On the other hand, such a
model may be intended to represent that the subject believes or knows that the
phenomenon under study exhibits symmetry. We defend the view that these are
fundamentally different things, even though the difference cannot be captured by
Bayesian belief models. In fact, the failure to distinguish between both
situations leads to Laplace's so-called Principle of Insufficient Reason, which
has been criticised

extensively in the literature, and which led to the rejection of Bayesian
methods in the nineteenth and early twentieth century, in favour of frequentist
approaches to probability.

We show that there are belief models (imprecise probability models, coherent
lower previsions) that generalise and include the more traditional Bayesian
models, but where this fundamental difference can be captured. This leads to two
notions of symmetry for such models: weak invariance (representing symmetry of
beliefs) and strong invariance (modelling beliefs of symmetry). We discuss
various mathematical as well as more philosophical aspects of these notions. We
also discuss a few examples to show the relevance of our findings both to
probabilistic modelling and to statistical inference.

E. Miranda, G. de Cooman and E. Quaeghebeur,
*The moment problem for finitely additive probabilities*. In: Uncertainty
and Intelligent Information Systems (B. Bouchon-Meunier, R.R. Yager, C. Marsala
and M. Rifqi, eds.), pp. 33--45. World Scientific, 2008.

Abstract: We study the moment problem for finitely additive probabilities and show that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions.

E. Miranda,
*A comparison of conditional coherence concepts for finite spaces. *In:
Foundations of Reasoning under Uncertainty. Studies on Fuzziness and Soft Computing,
vol. 249 (B. Bouchon-Meunier, L. Magdalena, M. Ojeda-Aciego, J.L. Verdegay, R.R.
Yager, eds.), pp. 223--246.
Springer, 2010.

Abstract: We compare the different notions
of conditional coherence within the behavioural theory of imprecise
probabilities when all the referential spaces are finite. We show that the
difference between weak and strong coherence comes from conditioning on sets of
(lower, and in some cases upper) probability zero. Next, we

characterise the range of coherent extensions, proving that the greatest
coherent extensions can always be calculated using the notion of regular
extension. Finally, we investigate which consistency conditions are preserved by
convex combinations point-wise limits, and whether it is possible to update a
coherent lower prevision while maintaining 2-monotonicity.

E. Miranda, A. Van Camp, G. de Cooman, Choice functions and rejection sets. In: The Mathematics of the Uncertain (E. Gil, E. Gil, J. Gil, M.A. Gil, eds.), pages 309-318. Springer, 2018

Abstract: We establish an equivalent representation of coherent choice functions in terms of a family of rejection sets, and investigate how each of the coherence axioms translates into this framework. In addition, we show that this family allows to simplify the verification of coherence in a number of particular cases.