Abstract: A precise aleatory statistical model can be constructed by starting with a null probability distribution on the observation space and expressing the log-likelihod ratios of other distributions as linear combinations from a set of basis functions. Conversely, any model all of whose members have the same null sets can be expressed in this way. An exhaustive model using only a finite number of basis functions constitutes an exponential family of distributions. A convex subset of this space of functions parametrises the distributions in the model. Starting with an arbitrary "prior" distribution on this parameter space, the dual space generates a family of distributions conjugate to the statistical model, parametrised by elements of the observation space. Bayesian updating of a prior distribution can be visualised as a translation by an update function. Walley's imprecise Dirichlet model (IDM) can be viewed as a generalisation of this paradigm in which there is not a single distinguished prior distribution, but an entire subfamily which is shifted by the updating process. Posterior upper and lower previsions are then defined as the upper and lower envelopes (respectively) of the set of posterior expectations based on this prior subspace. This principle, however, can be applied to any model. The crucial issue is the choice of the relevant subfamily that gives maximum imprecision a priori but whose posterior previsions are not vacuous. At issue, however, is the question of how to quantify the imprecision of a set of probability measures. While for exponential families such sets can be represented in Euclidean space, the Euclidean metric is not invariant under reparametrization and thus does not represent the intrinsic geometry of such sets. Kullback-Leibler information, however, provides a quasi-distance between probability measures. The imprecision of a set of measures could be quantified by its information radius. Several examples of exponential families will be presented, and the connection between such models and non-parametric predictive inference will be explored.
Abstract: The criteria that characterize many interesting classes of lower previsions, such as coherent or k-monotone lower probabilities, can in finite spaces often be seen as a set of linear constraints on the set of lower previsions in the class, which therefore is a convex polyhedron. It can be equivalently characterized by its set of vertices. For all interesting classes that I studied, the set of vertices or necessary and sufficient constraints is finite. In the presentation I aim to make these representations a bit more concrete to people, so that their possible uses -- both in applications and theory -- can be discussed in a tangible way.
Abstract: In imprecise Bayesian inference with conjugate priors as presented in (Walley 1991, Ch. 5.4.3) for Bernoulli data, and in the generalization to data from exponential family distributions in (Walter & Augustin 2009), the set of priors is characterized by an interval for the pseudocounts parameter n and an interval for the main interest parameter y (for y one-dimensional). This makes the description of the prior set very easy, and leads to simple updating rules with respect to prior-data conflict. Such rectangular prior sets may, however, for several reasons, not be a good representation of prior beliefs, as this set shape poses considerable constraints on the set of priors. I'd like to point to several issues arising with rectangular prior sets and explore some ideas about more flexible descriptions of parameter sets.
Abstract: Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of uncertainty, but different approaches to conditioning in that framework have been proposed in the past, leaving the matter unsettled. We propose here an approach to the conditioning of belief functions based on geometrically projecting them onto the simplex associated with the conditioning event in the space of all belief functions. Two different such simplices can be defined, as each belief function can be represented as either the vector of its basic probability values or the vector of its belief values. We show here that such a geometric approach to conditioning often produces simple results with straightforward interpretations in terms of degrees of belief. The question of whether classical approaches, such as for instance Dempster's conditioning, can also be reduced to some form of distance minimization remains open: the study of families of combination rules generated by (geometric) conditioning rules appears to be the natural prosecution of the presented research.