Probability boxes, also called p-boxes, correspond to sets of probability distributions bounded by a pair of distribution functions. They belong to the class of models known as imprecise ...probabilities. One of the central issues related to imprecise probabilities is the intervals of values corresponding to the expectations of random variables, and in particular the interval bounds. In general, these are reached at the extreme points of credal sets, which denote convex sets of compatible probabilistic models. The goal of this paper is to characterize and identify extreme points corresponding to p-boxes on finite domains. To achieve this, we use the concept of normal cones. In the context of imprecise probabilities, these correspond to sets of random variables whose extreme expectations are reached at a common extreme point. Our main results include a characterization of all possible normal cones of p-boxes, their relation to extreme points, and the identification of an adjacency structure on the collection of normal cones that is closely related to the adjacency structure in the set of extreme points.
A numerical approach to computing solutions of a generalized Kolmogorov backward equation is proposed, in which the stochastic matrix is replaced by a nonlinear operator obtained as the lower bound ...of a set of stochastic matrices. The equation is central to the theory of imprecise Markov chains in continuous time, which has made rapid progress in recent years. One of the obstacles to its implementation remains the high computational complexity, with the prevailing existing approaches relying on a discretization of the time interval. In order to achieve sufficient accuracy of the approximations, the grid must typically contain a large number of points on which optimization steps are performed, usually using linear programming. The main goal of this work is to develop a new, more efficient approach by significantly reducing the number of optimization steps required to achieve the prescribed accuracy of the solutions. Our approach is based on the Lipschitz continuity of the solutions of the equation with respect to time, which results in the optimization problems occurring at nearby points of the time interval having similar optimal solutions. This property is exploited using the theory of normal cones of convex polytopes. If the solution vectors remain within the same normal cone of a polytope corresponding to the nonlinear operator in a given interval, the optimization problem to be solved becomes linear, which allows much faster computations. This paper is primarily concerned with providing the theoretical basis for the new technique. However, initial tests show that it significantly outperforms existing methods in most cases.
Credal sets are one of the most important models for describing probabilistic uncertainty. They usually arise as convex sets of probabilistic models compatible with judgments provided in terms of ...coherent lower previsions or more specific models such as coherent lower probabilities or probability intervals. In finite spaces, credal sets usually take the form of convex polytopes. Many properties of convex polytopes can be derived from their normal cones, which form polyhedral complexes called normal fans. We analyze the properties of normal cones corresponding to credal sets of coherent lower probabilities. For two important classes of coherent lower probabilities, 2-monotone lower probabilities and probability intervals, we provide a detailed description of the normal fan structure. These structures are related to the structure of the extreme points of the credal sets. To arrive at our main results, we provide some general results on triangulated normal fans of convex polyhedra and their adjacency structure.
When the initial distribution and transition rates for a continuous time Markov chain are not known precisely, robust methods are needed to study the evolution of the process in time to avoid ...judgements based on unwarranted precision. We follow the ideas successfully applied in the study of discrete time model to build a framework of imprecise Markov chains in continuous time. The imprecision in the distributions over the set of states is modelled with upper and lower expectation functionals, which equivalently represent sets of probability distributions. Uncertainty in transitions is modelled with sets of transition rates compatible with available information. The Kolmogorov’s backward equation is then generalised into the form of a generalised differential equation, with generalised derivatives and set valued maps. The upper and lower expectation functionals corresponding to imprecise distributions at given times are determined by the maximal and minimal solutions of these equations. The second part of the paper is devoted to numerical methods for approximating the boundary solutions. The methods are based on discretisation of the time interval. A uniform and adaptive grid discretisations are examined. The latter is computationally much more efficient than the former one, but is not applicable on every interval. Therefore, to achieve maximal efficiency a combination of the methods is used.
Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision – even on finite ...underlying sample spaces – an infinite number of assessments is needed in general. Therefore, they are often only described approximately by some less general models, such as coherent lower probabilities or in terms of some other finite set of constraints. The magnitude of error induced by the approximations has often been neglected in the literature, despite the fact that it can be significant, with substantial impact on consequent decisions. An apparent reason is that no widely used general method for estimating the error seems to be available at the moment.
This paper provides a practically applicable method that allows calculating an upper bound for the maximal error induced by approximating a coherent lower probability with its values on a finite set of gambles. An algorithm is also provided with an estimation of its computational complexity.
The effect of perturbations of parameters for uniquely convergent imprecise Markov chains is studied. We provide the maximal distance between the distributions of original and perturbed chain and ...maximal degree of imprecision, given the imprecision of the initial distribution. The bounds on the errors and degrees of imprecision are found for the distributions at finite time steps, and for the stationary distributions as well.
We propose a model of random walks on weighted graphs where the weights are interval valued, and connect it to reversible imprecise Markov chains. While the theory of imprecise Markov chains is now ...well established, this is a first attempt to model reversible chains. In contrast with the existing theory, the probability models that have to be considered are now non-convex. This presents a difficulty in computational sense, since convexity is critical for the existence of efficient optimization algorithms used in the existing models. The second part of the paper therefore addresses the computational issues of the model. The goal is finding sets of weights which maximize or minimize expectations corresponding to multiple steps transition probabilities. In particular, we present a local optimization algorithm and numerically test its efficiency. We show that its application allows finding close approximations of the globally best solutions in reasonable time.
•Random walks on graphs with interval weights are studied.•The model is connected to reversible imprecise Markov chains.•A local optimization algorithm is proposed.•The results of numerical tests of the local optimization algorithm are reported.
Blockmodeling linked networks aims to simultaneously cluster two or more sets of units into clusters based on a network where ties are possible both between units from the same set as well as between ...units of different sets. While this has already been developed for generalized and k-means blockmodeling, our approach is based on the well-known stochastic blockmodeling technique, utilizing a mixture model. Estimation is performed using the CEM algorithm, which iteratively estimates the parameters by maximizing a suitable likelihood function and reclusters the units according to the parameters. The steps are repeated until the likelihood function ceases to improve.
A key drawback of the basic algorithm is that it treats all units equally, consequently yielding higher influence to larger parts of the data. The greater size, however, does not necessarily imply higher importance. To mitigate this asymmetry, we propose a solution where underrepresented parts of the data are given more influence through an appropriate weighting. This idea leads to the so-called weighted likelihood approach, where the ordinary likelihood function is replaced by a weighted likelihood.
The efficiency of the different approaches is tested via simulations. It is shown through simulations that the weighted likelihood approach performs better for larger networks and a clearer blockmodel structure, especially when the one-mode blockmodels within the smaller sets are clearer.
•Linked networks contain two or more sets of units and subnetworks.•Subnetworks contain ties among the units of one set or between units of two sets.•Examples of linked networks are also dynamic networks and multilevel networks.•Blockmodeling linked networks jointly partitions all sets of units.•A stochastic blockmodeling approach is utilized to blockmodeling linked networks.•Weighted likelihood is used to balance the impact of different subnetworks.
The omnipotence of copulas when modeling dependence given marginal distributions in a multivariate stochastic situation is assured by the Sklar's theorem. Montes et al. (2015) suggest the notion of ...what they call an imprecise copula that brings some of its power in bivariate case to the imprecise setting. When there is imprecision about the marginals, one can model the available information by means of p-boxes, that are pairs of ordered distribution functions. By analogy they introduce pairs of bivariate functions satisfying certain conditions. In this paper we introduce the imprecise versions of some classes of copulas emerging from shock models that are important in applications. The so obtained pairs of functions are not only imprecise copulas but satisfy an even stronger condition. The fact that this condition really is stronger is shown in Omladič and Stopar (2019) thus raising the importance of our results. The main technical difficulty in developing our imprecise copulas lies in introducing an appropriate stochastic order on these bivariate objects.
Although bivariate imprecise copulas have recently attracted substantial attention, the multivariate case seems still to be open. So, it is natural to test it first on shock model induced copulas, a ...family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation in the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to exhibit their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. Also, the theory of imprecise RMM copulas seems to be new even in the bivariate case. It turns out that all imprecise copulas under consideration are coherent.
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