Imprecise stochastic processes in discrete time: global models,imprecise Markov chains,and ergodic theorems

We justify and discuss expressions for joint lower and upper expectations in imprecise probability trees, in terms of the sub- and supermartingales that can be associated with such trees. These imprecise probability trees can be seen as discrete-time stochastic processes with finite state sets and transition probabilities that are imprecise, in the sense that they are only known to belong to some convex closed set of probability measures. We derive various properties for their joint lower and upper expectations, and in particular a law of iterated expectations. We then focus on the special case of imprecise Markov chains, investigate their Markov and stationarity properties, and use these, by way of an example, to derive a system of non-linear equations for lower and upper expected transition and return times. Most importantly, we prove a game-theoretic version of the strong law of large numbers for submartingale differences in imprecise probability trees, and use this to derive point-wise ergodic theorems for imprecise Markov chains.     

Discrete time Markov chains with interval probabilities

The parameters of Markov chain models are often not known precisely. Instead of ignoring this problem, a better way to cope with it is to incorporate the imprecision into the models. This has become possible with the development of models of imprecise probabilities, such as the interval probability model. In this paper we discuss some modelling approaches which range from simple probability intervals to the general interval probability models and further to the models allowing completely general convex sets of probabilities. The basic idea is that precisely known initial distributions and transition matrices are replaced by imprecise ones, which effectively means that sets of possible candidates are considered. Consequently, sets of possible results are obtained and represented using similar imprecise probability models.We first set up the model and then show how to perform calculations of the distributions corresponding to the consecutive steps of a Markov chain. We present several approaches to such calculations and compare them with respect to the accuracy of the results. Next we consider a generalisation of the concept of regularity and study the convergence of regular imprecise Markov chains. We also give some numerical examples to compare different approaches to calculations of the sets of probabilities.     

Boole''s conditions of possible experience and reasoning under uncertainty

Consider a set of logical sentences together with probabilities that they are true. These probabilities must satisfy certain conditions for this system to be consistent. It is shown that an analytical form of these conditions can be obtained by enumerating the extreme rays of a polyhedron. We also consider the cases when (i) intervals of probabilities are given, instead of single values; and (ii) best lower and upper bounds on the probability of an additional logical sentence to be true are sought. Enumeration of vertices and extreme rays is used. Each vertex defines a finear expression and the maximum (minimum) of these defines a best possible lower (upper) bound on the probability of the additional logical sentence to be true. Each extreme ray leads to a constraint on the probabilities assigned to the initial set of logical sentences. Redundancy in these expressions is studied. Illustrations are provided in the domain of reasoning under uncertainty.     

Random sets as imprecise random variables

Given a random set coming from the imprecise observation of a random variable, we study how to model the information about the probability distribution of this random variable. Specifically, we investigate whether the information given by the upper and lower probabilities induced by the random set is equivalent to the one given by the class of the probabilities induced by the measurable selections; together with sufficient conditions for this, we also give examples showing that they are not equivalent in all cases.     

Sets of desirable gambles: Conditioning, representation, and precise probabilities

The theory of sets of desirable gambles is a very general model which covers most of the existing theories for imprecise probability as special cases; it has a clear and simple axiomatic justification; and mathematical definitions are natural and intuitive. However, much work remains to be done until the theory of desirable gambles can be considered as generally applicable to reasoning tasks as other approaches to imprecise probability are. This paper gives an overview of some of the fundamental concepts for reasoning with uncertainty expressed in terms of desirable gambles in the finite case, provides a characterization of regular extension, and studies the nature of maximally coherent sets of desirable gambles, which correspond to finite sequences of probability distributions, each one of them defined on the set where the previous one assigns probability zero.     

Independence and 2-monotonicity: Nice to have,hard to keep

In imprecise probability theories, independence modeling and computational tractability are two important issues. The former is essential to work with multiple variables and multivariate spaces, while the latter is essential in practical applications. When using lower probabilities to model uncertainty about the value assumed by a variable, satisfying the property of 2-monotonicity decreases the computational burden of inference, hence answering the latter issue. In a first part, this paper investigates whether the joint uncertainty obtained by main existing notions of independence preserve the 2-monotonicity of marginal models. It is shown that it is usually not the case, except for the formal extension of random set independence to 2-monotone lower probabilities. The second part of the paper explores the properties and interests of this extension within the setting of lower probabilities.     

Warp effects on calculating interval probabilities

In real-life decision analysis, the probabilities and utilities of consequences are in general vague and imprecise. One way to model imprecise probabilities is to represent a probability with the interval between the lowest possible and the highest possible probability, respectively. However, there are disadvantages with this approach; one being that when an event has several possible outcomes, the distributions of belief in the different probabilities are heavily concentrated toward their centres of mass, meaning that much of the information of the original intervals are lost. Representing an imprecise probability with the distribution’s centre of mass therefore in practice gives much the same result as using an interval, but a single number instead of an interval is computationally easier and avoids problems such as overlapping intervals. We demonstrate why second-order calculations add information when handling imprecise representations, as is the case of decision trees or probabilistic networks. We suggest a measure of belief density for such intervals. We also discuss properties applicable to general distributions. The results herein apply also to approaches which do not explicitly deal with second-order distributions, instead using only first-order concepts such as upper and lower bounds.     

Method of resurgent analysis in atomic collision theory

We consider the atomic collision problem in the adiabatic approximation. We show that the transition probabilities can be evaluated in this approximation using the tools of resurgent analysis. We suggest a computational algorithm for the transition probabilities and give the mathematical foundation of this algorithm. The analysis is carried out using the example of the two-level Landau-Zener model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 2, pp. 308–323, August, 1997.     

Partial Moment Entropy Approximation to Radiative Heat Transfer

A new approximation to Radiative Heat Transfer, intermediate betweeen moment models and discrete ordinates, is introduced. We consider a moment system with general partitions of velocity space, closed by an entropy minimization principle. We give physical and mathematical reasons for this choice of model and study its properties. A numerical example is presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Combined analysis of unique and repetitive events in quantitative risk assessment

For risk assessment to be a relevant tool in the study of any type of system or activity, it needs to be based on a framework that allows for jointly analyzing both unique and repetitive events. Separately, unique events may be handled by predictive probability assignments on the events, and repetitive events with unknown/uncertain frequencies are typically handled by the probability of frequency (or Bayesian) approach. Regardless of the nature of the events involved, there may be a problem with imprecision in the probability assignments. Several uncertainty representations with the interpretation of lower and upper probability have been developed for reflecting such imprecision. In particular, several methods exist for jointly propagating precise and imprecise probabilistic input in the probability of frequency setting. In the present position paper we outline a framework for the combined analysis of unique and repetitive events in quantitative risk assessment using both precise and imprecise probability. In particular, we extend an existing method for jointly propagating probabilistic and possibilistic input by relaxing the assumption that all events involved have frequentist probabilities; instead we assume that frequentist probabilities may be introduced for some but not all events involved, i.e. some events are assumed to be unique and require predictive – possibly imprecise – probabilistic assignments, i.e. subjective probability assignments on the unique events without introducing underlying frequentist probabilities for these. A numerical example related to environmental risk assessment of the drilling of an oil well is included to illustrate the application of the resulting method.     

Decision-dependent probabilities in stochastic programs with recourse

Stochastic programming with recourse usually assumes uncertainty to be exogenous. Our work presents modelling and application of decision-dependent uncertainty in mathematical programming including a taxonomy of stochastic programming recourse models with decision-dependent uncertainty. The work includes several ways of incorporating direct or indirect manipulation of underlying probability distributions through decision variables in two-stage stochastic programming problems. Two-stage models are formulated where prior probabilities are distorted through an affine transformation or combined using a convex combination of several probability distributions. Additionally, we present models where the parameters of the probability distribution are first-stage decision variables. The probability distributions are either incorporated in the model using the exact expression or by using a rational approximation. Test instances for each formulation are solved with a commercial solver, BARON, using selective branching.     

Determining lower and upper bounds on probabilities of atomic propositions in sets of logical formulas represented by digraphs

In this paper we consider the problem of determining lower and upper bounds on probabilities of atomic propositions in sets of logical formulas represented by digraphs. We establish a sharp upper bound, as well as a lower bound that is not in general sharp. We show further that under a certain condition the lower bound is sharp. In that case, we obtain a closed form solution for the possible probabilities of the atomic propositions.The second author is partially supported by ONR grant N00014-92-J-1028 and AFOSR grant 91-0287.     

信度理论的扩张与一个审定损害的专家系统

通过使用集合的基本概率定义出上限、下限概率,形成信度;通过隶属函数定义模糊集合的包含度、相交度,使信度理论在模糊集合得以扩张,得到了利用不确定性及模糊性的一个合理的推理方法。在此基础上,采AND/OR/COMB树推理开发了一个审定损害的专家系统。     

An uncertainty interchange format with imprecise probabilities

This paper addresses the problem of exchanging uncertainty assessments in multi-agent systems. Since it is assumed that each agent might completely ignore the internal representation of its partners, a common interchange format is needed. We analyze the case of an interchange format defined by means of imprecise probabilities, pointing out the reasons of this choice. A core problem with the interchange format concerns transformations from imprecise probabilities into other formalisms (in particular, precise probabilities, possibilities, belief functions). We discuss this so far little investigated question, analyzing how previous proposals, mostly regarding special instances of imprecise probabilities, would fit into this problem. We then propose some general transformation procedures, which take also account of the fact that information can be partial, i.e. may concern an arbitrary (finite) set of events.     

Imprecise Markov chains with absorption

We consider convergence of Markov chains with uncertain parameters, known as imprecise Markov chains, which contain an absorbing state. We prove that under conditioning on non-absorption the imprecise conditional probabilities converge independently of the initial imprecise probability distribution if some regularity conditions are assumed. This is a generalisation of a known result from the classical theory of Markov chains by Darroch and Seneta [6].     

Bayesian estimation of finite time ruin probabilities

In this paper, we consider Bayesian inference and estimation of finite time ruin probabilities for the Sparre Andersen risk model. The dense family of Coxian distributions is considered for the approximation of both the inter‐claim time and claim size distributions. We illustrate that the Coxian model can be well fitted to real, long‐tailed claims data and that this compares well with the generalized Pareto model. The main advantage of using the Coxian model for inter‐claim times and claim sizes is that it is possible to compute finite time ruin probabilities making use of recent results from queueing theory. In practice, finite time ruin probabilities are much more useful than infinite time ruin probabilities as insurance companies are usually interested in predictions for short periods of future time and not just in the limit. We show how to obtain predictive distributions of these finite time ruin probabilities, which are more informative than simple point estimations and take account of model and parameter uncertainty. We illustrate the procedure with simulated data and the well‐known Danish fire loss data set. Copyright © 2009 John Wiley & Sons, Ltd.     

A tree augmented classifier based on Extreme Imprecise Dirichlet Model

We present TANC, a TAN classifier (tree-augmented naive) based on imprecise probabilities. TANC models prior near-ignorance via the Extreme Imprecise Dirichlet Model (EDM). A first contribution of this paper is the experimental comparison between EDM and the global Imprecise Dirichlet Model using the naive credal classifier (NCC), with the aim of showing that EDM is a sensible approximation of the global IDM. TANC is able to deal with missing data in a conservative manner by considering all possible completions (without assuming them to be missing-at-random), but avoiding an exponential increase of the computational time. By experiments on real data sets, we show that TANC is more reliable than the Bayesian TAN and that it provides better performance compared to previous TANs based on imprecise probabilities. Yet, TANC is sometimes outperformed by NCC because the learned TAN structures are too complex; this calls for novel algorithms for learning the TAN structures, better suited for an imprecise probability classifier.     

Comments on “Statistical reasoning with set-valued information: Ontic vs. epistemic view” by Inés Couso and Didier Dubois

I discuss some aspects of the distinction between ontic and epistemic views of sets as representation of imprecise or incomplete information. In particular, I consider its implications on imprecise probability representations: credal sets and sets of desirable gambles. It is emphasized that the interpretation of the same mathematical object can be different depending on the point of view from which this element is considered. In the case of a fuzzy information on a random variable, it is possible to define a possibility distribution on the simplex of probability distributions. I add some comments about the properties of this possibility distribution.     

Reliability analysis for Weibull distribution with homogeneous heavily censored data based on Bayesian and least-squares methods

The reliability for Weibull distribution with homogeneous heavily censored data is analyzed in this study. The universal model of heavily censored data and existing methods, including maximum likelihood, least-squares, E-Bayesian estimation, and hierarchical Bayesian methods, are introduced. An improved method is proposed based on Bayesian inference and least-squares method. In this method, the Bayes estimations of failure probabilities are focused on for all the samples. The conjugate prior distribution of failure probability is set, and an optimization model is developed by maximizing the information entropy of prior distribution to determine the hyper-parameters. By integrating the likelihood function, the posterior distribution of failure probability is then derived to yield the Bayes estimation of failure probability. The estimations of reliability parameters are obtained by fitting distribution curve using least-squares method. The four existing methods are compared with the proposed method in terms of applicability, precision, efficiency, robustness, and simplicity. Specifically, the closed form expressions concerning E-Bayesian estimation and hierarchical Bayesian methods are derived and used. The comparisons demonstrate that the improved method is superior. Finally, three illustrative examples are presented to show the application of the proposed method.     

2.3. Methodology of OR

We present and analyze a generalization of the standard decision analysis model of sequential decisionmaking under risk. The decision tree is assumed given and all probabilities are assumed to be known precisely. Utility values are assumed to be affine in an imprecisely known parameter. The affine form is sufficiently general to allow importance weights or the utility values themselves to be represented by the imprecise parameter. Parameter imprecision is described by set inclusion. A relation on all available alternatives is assumed given for each decision node. The intent of each (not necessarily complete) relation is to model the decisionmaker's directly expressed preferences among the available alternatives at the associated decision node. A numerical procedure is developed to determine the set of all strategies that may be optimal and the corresponding set of all possible parameter values. An example illustrates the procedure.