Characterizing the set of coherent lower previsions with a finite number of constraints or vertices

The standard coherence criterion for lower previsions is expressed using an infinite number of linear constraints. For lower previsions that are essentially defined on some finite set of gambles on a finite possibility space, we present a reformulation of this criterion that only uses a finite number of constraints. Any such lower prevision is coherent if it lies within the convex polytope defined by these constraints. The vertices of this polytope are the extreme coherent lower previsions for the given set of gambles. Our reformulation makes it possible to compute them. We show how this is done and illustrate the procedure and its results

Characterizing the set of coherent lower previsions with a finite number of constraints or vertices

The standard coherence criterion for lower previsions is expressed using an infinite number of linear constraints. For lower previsions that are essentially defined on some finite set of gambles on a finite possibility space, we present a reformulation of this criterion that only uses a finite number of constraints. Any such lower prevision is coherent if it lies within the convex polytope defined by these constraints. The vertices of this polytope are the extreme coherent lower previsions for the given set of gambles. Our reformulation makes it possible to compute them. We show how this is done and illustrate the procedure and its results

Lower and upper covariance

We give a definition for lower and upper covariance in Walley's theory of imprecise probabilities (or coherent lower previsions) that is direct, i.e., does not refer to credal sets. It generalizes Walley's definition for lower and upper variance. Just like Walley's definition of lower and upper variance, our definition for lower and upper covariance is compatible with the credal set approach; i.e., we also provide a covariance envelope theorem. Our approach mirrors the one taken by Walley: we first reformulate the calculation of a covariance as an optimization problem and then generalize this optimization problem to lower and upper previsions. We also briefly discuss the still unclear meaning of lower and upper (co)variances and mention some ideas about generalizations to other central moments

Game-theoretic learning using the imprecise Dirichlet model

We discuss two approaches for choosing a strategy in a two-player game. We suppose that the game is played a large number of rounds, which allows the players to use observations of past play to guide them in choosing a strategy. Central in these approaches is the way the opponent's next strategy is assessed; both a precise and an imprecise Dirichlet model are used. The observations of the opponent's past strategies can then be used to update the model and obtain new assessments. To some extent, the imprecise probability approach allows us to avoid making arbitrary initial assessments. To be able to choose a strategy, the assessment of the opponent's strategy is combined with rules for selecting an optimal response to it: a so-called best response or a maximin strategy. Together with the updating procedure, this allows us to choose strategies for all the rounds of the game. The resulting playing sequence can then be analysed to investigate if the strategy choices can converge to equilibria

Characterizing coherence, correcting incoherence

Lower previsions defined on a finite set of gambles can be looked at as points in a finite-dimensional real vector space. Within that vector space, the sets of sure loss avoiding and coherent lower previsions form convex polyhedra. We present procedures for obtaining characterizations of these polyhedra in terms of a minimal, finite number of linear constraints. As compared to the previously known procedure, these procedures are more efficient and much more straightforward. Next, we take a look at a procedure for correcting incoherent lower previsions based on pointwise dominance. This procedure can be formulated as a multi-objective linear program, and the availability of the finite characterizations provide an avenue for making these programs computationally feasible

Exchangeability for sets of desirable gambles

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We study exchangeability assessments for such models, and prove a counterpart of de Finetti's finite representation theorem. We show that this representation theorem has a very nice geometrical interpretation. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability

Imprecise probability models for inference in exponential families

When considering sampling models described by a distribution from an exponential family, it is possible to create two types of imprecise probability models. One is based on the corresponding conjugate distribution and the other on the corresponding predictive distribution. In this paper, we show how these types of models can be constructed for any (regular, linear, canonical) exponential family, such as the centered normal distribution. To illustrate the possible use of such models, we take a look at credal classification. We show that they are very natural and potentially promising candidates for describing the attributes of a credal classifier, also in the case of continuous attributes

Command line completion: an illustration of learning and decision making using the imprecise Dirichlet model

A method of command line completion based on probabilistic models is described. The method supplements the existing deterministic ones. The probabilistic models are developed within the context of imprecise probabilities. An imprecise Dirichlet model is used to represent the assessments about all possible completions and to allow for learning by observing the commands typed previously. Due to the use of imprecise probabilities a partial (instead of a linear) ordering of the possible completion actions will be constructed during decision making. Markov models can additionally be incorporated to take recurring sequences of commands into account