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

Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

Tractable approximate deduction for OWL

Acknowledgements This work has been partially supported by the European project Marrying Ontologies and Software Technologies (EU ICT2008-216691), the European project Knowledge Driven Data Exploitation (EU FP7/IAPP2011-286348), the UK EPSRC project WhatIf (EP/J014354/1). The authors thank Prof. Ian Horrocks and Dr. Giorgos Stoilos for their helpful discussion on role subsumptions. The authors thank Rafael S. Gonçalves et al. for providing their hotspots ontologies. The authors also thank BoC-group for providing their ADOxx Metamodelling ontologies.Peer reviewedPostprin

Uncertain inference using interval probability theory

AbstractThe use of interval probability theory (IPT) for uncertain inference is demonstrated. The general inference rule adopted is the theorem of total probability. This enables information on the relevance of the elements of the power set of evidence to be combined with the measures of the support for and dependence between each item of evidence. The approach recognises the importance of the structure of inference problems and yet is an open world theory in which the domain need not be completely specified in order to obtain meaningful inferences. IPT is used to manipulate conflicting evidence and to merge evidence on the dependability of a process with the data handled by that process. Uncertain inference using IPT is compared with Bayesian inference

Other uncertainty theories based on capacities

International audienceThe two main uncertainty representations in the literature that tolerate imprecision are possibility distributions and random disjunctive sets. This chapter devotes special attention to the theories that have emerged from them. The first part of the chapter discusses epistemic logic and derives the need for capturing imprecision in information representations. It bridges the gap between uncertainty theories and epistemic logic showing that imprecise probabilities subsume modalities of possibility and necessity as much as probability. The second part presents possibility and evidence theories, their origins, assumptions and semantics, discusses the connections between them and the general framework of imprecise probability. Finally, chapter points out the remaining discrepancies between the different theories regarding various basic notions, such as conditioning, independence or information fusion and the existing bridges between them

Partially Identified Prevalence Estimation under Misclassification using the Kappa Coefficient

We discuss a new strategy for prevalence estimation in the presence of misclassification. Our method is applicable when misclassification probabilities are unknown but independent replicate measurements are available. This yields the kappa coefficient, which indicates the agreement between the two measurements. From this information, a direct correction for misclassification is not feasible due to non-identifiability. However, it is possible to derive estimation intervals relying on the concept of partial identification. These intervals give interesting insights into possible bias due to misclassification. Furthermore, confidence intervals can be constructed. Our method is illustrated in several theoretical scenarios and in an example from oral health, where prevalence estimation of caries in children is the issue