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

Epistemic irrelevance in credal nets: the case of imprecise Markov trees

We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithm's operation, and report an application to on-line character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.Comment: 29 pages, 5 figures, 1 tabl

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

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Epistemic irrelevance in credal networks : the case of imprecise Markov trees

We replace strong independence in credal networks with the weaker notion of epistemic irrelevance. Focusing on directed trees, we show how to combine local credal sets into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is essentially linear in the number of nodes, is formulated entirely in terms of coherent lower previsions. We supply examples of the algorithm's operation, and report an application to on-line character recognition that illustrates the advantages of our model for prediction

A General Framework for Updating Belief Distributions

We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered under the special case of using self information loss. Modern application areas make it is increasingly challenging for Bayesians to attempt to model the true data generating mechanism. Moreover, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our proposed framework uses loss-functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known, yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.Comment: This is the pre-peer reviewed version of the article "A General Framework for Updating Belief Distributions", which has been accepted for publication in the Journal of Statistical Society - Series B. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archivin

Bayesian Learning for a Class of Priors with Prescribed Marginals

We present Bayesian updating of an imprecise probability measure, represented by a class of precise multidimensional probability measures. Choice and analysis of our class are motivated by expert interviews that we conducted with modelers in the context of climatic change. From the interviews we deduce that generically, experts hold a much more informed opinion on the marginals of uncertain parameters rather than on their correlations. Accordingly, we specify the class by prescribing precise measures for the marginals while letting the correlation structure subject to complete ignorance. For sake of transparency, our discussion focuses on the tutorial example of a linear two-dimensional Gaussian model. We operationalize Bayesian learning for that class by various updating rules, starting with (a modified version of) the generalized Bayes' rule and the maximum likelihood update rule (after Gilboa and Schmeidler). Over a large range of potential observations, the generalized Bayes' rule would provide non-informative results. We restrict this counter-intuitive and unnecessary growth of uncertainty by two means, the discussion of which refers to any kind of imprecise model, not only to our class. First, we find our class of priors too inclusive and, hence, require certain additional properties of prior measures in terms of smoothness of probability density functions. Second, we argue that both updating rules are dissatisfying, the generalized Bayes' rule being too conservative, i.e., too inclusive, the maximum likelihood rule being too exclusive. Instead, we introduce two new ways of Bayesian updating of imprecise probabilities: a ``weighted maximum likelihood method'' and a ``semi-classical method.'' The former bases Bayesian updating on the whole set of priors, however, with weighted influence of its members. By referring to the whole set, the weighted maximum likelihood method allows for more robust inferences than the standard maximum likelihood method and, hence, is better to justify than the latter.Furthermore, the semi-classical method is more objective than the weighted maximum likelihood method as it does not require the subjective definition of a weighting function. Both new methods reveal much more informative results than the generalized Bayes' rule, what we demonstrate for the example of a stylized insurance model

Credal Networks under Epistemic Irrelevance

A credal network under epistemic irrelevance is a generalised type of Bayesian network that relaxes its two main building blocks. On the one hand, the local probabilities are allowed to be partially specified. On the other hand, the assessments of independence do not have to hold exactly. Conceptually, these two features turn credal networks under epistemic irrelevance into a powerful alternative to Bayesian networks, offering a more flexible approach to graph-based multivariate uncertainty modelling. However, in practice, they have long been perceived as very hard to work with, both theoretically and computationally. The aim of this paper is to demonstrate that this perception is no longer justified. We provide a general introduction to credal networks under epistemic irrelevance, give an overview of the state of the art, and present several new theoretical results. Most importantly, we explain how these results can be combined to allow for the design of recursive inference methods. We provide numerous concrete examples of how this can be achieved, and use these to demonstrate that computing with credal networks under epistemic irrelevance is most definitely feasible, and in some cases even highly efficient. We also discuss several philosophical aspects, including the lack of symmetry, how to deal with probability zero, the interpretation of lower expectations, the axiomatic status of graphoid properties, and the difference between updating and conditioning

Coherent frequentism

By representing the range of fair betting odds according to a pair of confidence set estimators, dual probability measures on parameter space called frequentist posteriors secure the coherence of subjective inference without any prior distribution. The closure of the set of expected losses corresponding to the dual frequentist posteriors constrains decisions without arbitrarily forcing optimization under all circumstances. This decision theory reduces to those that maximize expected utility when the pair of frequentist posteriors is induced by an exact or approximate confidence set estimator or when an automatic reduction rule is applied to the pair. In such cases, the resulting frequentist posterior is coherent in the sense that, as a probability distribution of the parameter of interest, it satisfies the axioms of the decision-theoretic and logic-theoretic systems typically cited in support of the Bayesian posterior. Unlike the p-value, the confidence level of an interval hypothesis derived from such a measure is suitable as an estimator of the indicator of hypothesis truth since it converges in sample-space probability to 1 if the hypothesis is true or to 0 otherwise under general conditions.Comment: The confidence-measure theory of inference and decision is explicitly extended to vector parameters of interest. The derivation of upper and lower confidence levels from valid and nonconservative set estimators is formalize