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Updating beliefs with incomplete observations

机译：用不完整的观察更新信念

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Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete (or set-valued). 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 (updated) probabilities. In general, this new approach to updating produces lower and upper posterior probabilities and previsions (expectations), as well as partially determinate decisions. This is a logical consequence of the existing ignorance about the incompleteness mechanism. As an example, we use the new updating method to properly address the apparent paradox in the 'Monty Hall' puzzle. More importantly, we apply it 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 to compare classes 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. (C) 2004 Elsevier B.V. All rights reserved.
机译：当前，人们对沙夫在1985年提出的问题产生了新的兴趣，即当观测不完整（或设定值）时更新概率。通常，这是一个基本问题，对于贝叶斯网络尤其重要。最近，Grunwald和Halpern表明，在非常特殊的假设下，这种情况下常用的更新策略会失败。在本文中，我们提出了一种使用不完整观测值更新概率的新方法。我们的方法是故意保守的：我们不对所谓的不完整机制做出假设，该不完整机制将完整与不完整的观察联系在一起。我们通过空虚的较低准度（一种来自不精确概率理论的工具）对这种机制的无知进行建模，并且仅使用一致性参数将先验概率转化为后验（更新）概率。通常，这种新的更新方法会产生较高和较低的后验概率和预测（期望），以及部分确定的决策。这是对不完整机制的现有无知的逻辑结果。例如，我们使用新的更新方法来正确解决“蒙蒂·霍尔”难题中的明显悖论。更重要的是，我们将其应用于概率专家系统中新证据的分类问题，从而导致了新的，所谓的保守更新规则。在使用专家知识构造的贝叶斯网络的特殊情况下，我们提供了一种精确的算法来根据我们的更新规则比较类，对于比polytree宽的一类网络，它具有线性时间复杂度。然后将此结果扩展到更广泛的credal网络框架，在该框架中，计算通常比使用贝叶斯网络困难得多。通过一个示例，我们表明，当没有充分的不完整机制假设是合理的时，我们的规则似乎为不完整观察的可靠更新提供了坚实的基础。 （C）2004 Elsevier B.V.保留所有权利。

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