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.