Updating beliefs with incomplete observations

Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete. This is a fundamental problem in general, and of particular interest for Bayesian networks. Recently, Grunwald and Halpern have shown that commonly used updating strategies fail in this case, except under very special assumptions. In this paper we propose a new method for updating probabilities with incomplete observations. Our approach is deliberately conservative: we make no assumptions about the so-called incompleteness mechanism that associates complete with incomplete observations. We model our ignorance about this mechanism by a vacuous lower prevision, a tool from the theory of imprecise probabilities, and we use only coherence arguments to turn prior into posterior probabilities. In general, this new approach to updating produces lower and upper posterior probabilities and expectations, as well as partially determinate decisions. This is a logical consequence of the existing ignorance about the incompleteness mechanism. We apply the new approach to the problem of classification of new evidence in probabilistic expert systems, where it leads to a new, so-called conservative updating rule. In the special case of Bayesian networks constructed using expert knowledge, we provide an exact algorithm for classification based on our updating rule, which has linear-time complexity for a class of networks wider than polytrees. This result is then extended to the more general framework of credal networks, where computations are often much harder than with Bayesian nets. Using an example, we show that our rule appears to provide a solid basis for reliable updating with incomplete observations, when no strong assumptions about the incompleteness mechanism are justified.Comment: Replaced with extended versio

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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How much of commonsense and legal reasoning is formalizable? A review of conceptual obstacles

Fifty years of effort in artificial intelligence (AI) and the formalization of legal reasoning have produced both successes and failures. Considerable success in organizing and displaying evidence and its interrelationships has been accompanied by failure to achieve the original ambition of AI as applied to law: fully automated legal decision-making. The obstacles to formalizing legal reasoning have proved to be the same ones that make the formalization of commonsense reasoning so difficult, and are most evident where legal reasoning has to meld with the vast web of ordinary human knowledge of the world. Underlying many of the problems is the mismatch between the discreteness of symbol manipulation and the continuous nature of imprecise natural language, of degrees of similarity and analogy, and of probabilities

Bayesian nonparametric system reliability using sets of priors

An imprecise Bayesian nonparametric approach to system reliability with multiple types of components is developed. This allows modelling partial or imperfect prior knowledge on component failure distributions in a flexible way through bounds on the functioning probability. Given component level test data these bounds are propagated to bounds on the posterior predictive distribution for the functioning probability of a new system containing components exchangeable with those used in testing. The method further enables identification of prior–data conflict at the system level based on component level test data. New results on first-order stochastic dominance for the Beta-Binomial distribution make the technique computationally tractable. Our methodological contributions can be immediately used in applications by reliability practitioners as we provide easy to use software tools

A robust Bayesian land use model for crop rotations

Often, in dynamical systems, such as farmers’ crop choices, the dynamics are driven by external non-stationary factors, such as rainfall and agricultural input and output prices. Such dynamics can be modelled by a non-stationary stochastic process, where the transition probabilities are functions of such external factors. We propose using a multinomial logit model for these transition probabilities, and investigate the problem of estimating the parameters of this model from data. We adapt the work of Chen and Ibrahim to propose a conjugate prior distribution for the parameters of the multinomial logit model. Inspired by the imprecise Dirichlet model, we will perform a robust Bayesian analysis by proposing a fairly broad class of prior distributions, in order to accommodate scarcity of data and lack of strong prior expert opinion. We discuss the computation of bounds for the posterior transition probabilities, using a variety of calculation methods. These sets of posterior transition probabilities mean that our land use model consists of a non-stationary imprecise stochastic process. We discuss computation of future events in this process. Finally, we use our novel land use model to investigate real-world data. We investigate the impact of external variables on the posterior transition probabilities, and investigate a scenario for future crop growth. We also use our model to solve a hypothetical yet realistic policy problem