可信判决图中的易处理推理
Tractable Inference in Credal Sentential Decision Diagrams
摘要：
概率决策图是逻辑电路，其中选通门的输入由概率值注释。它们允许在布尔变量集上定义的联合概率质量函数的紧凑表示，这也与电路定义的逻辑约束一致。这种模型中的概率通常是从一组观察中学习的。当数据稀缺、不可靠或相互冲突时，这会导致过度自信和先验依赖的推断。在这项工作中，我们开发的credal的概率决策图，其概率对应的概括，允许取代本地概率（所谓的credal）集的质量函数。这些模型在布尔变量集上引入了一个联合信用集，该联合信用集将概率零急剧分配给与逻辑约束不一致的状态。三个推理算法推导出这些模型，这些允许计算：（i）下和上概率的观察任意数量的变量，（ii）下和上的条件概率的状态的一个单一的变量的观察给定的;（三）是否所有与信用规范兼容的概率决策图都具有相同的最可能解释给定一组变量，给定对其他变量的观察。这些推论是易于处理的，因为所有三个算法，基于自底向上遍历与局部线性规划任务的析取门，可以解决在多项式时间相对于电路的大小。对于第一个经验验证，我们考虑一个简单的应用程序的基础上嘈杂的七段显示器图像。观察到的credal模型，以正确区分容易和难以检测的实例，并优于其他生成模型不能应付逻辑约束。
英文摘要：
Probabilistic sentential decision diagrams are logic circuits where the inputs of disjunctive gates are annotated by probability values. They allow for a compact representation of joint probability mass functions defined over sets of Boolean variables, that are also consistent with the logical constraints defined by the circuit. The probabilities in such a model are usually learned from a set of observations. This leads to overconfident and prior-dependent inferences when data are scarce, unreliable or conflicting. In this work, we develop the credal sentential decision diagrams, a generalisation of their probabilistic counterpart that allows for replacing the local probabilities with (so-called credal) sets of mass functions. These models induce a joint credal set over the set of Boolean variables, that sharply assigns probability zero to states inconsistent with the logical constraints. Three inference algorithms are derived for these models, these allow to compute: (i) the lower and upper probabilities of an observation for an arbitrary number of variables; (ii) the lower and upper conditional probabilities for the state of a single variable given an observation; (iii) whether or not all the probabilistic sentential decision diagrams compatible with the credal specification have the same most probable explanation of a given set of variables given an observation of the other variables. These inferences are tractable, as all the three algorithms, based on bottom-up traversal with local linear programming tasks on the disjunctive gates, can be solved in polynomial time with respect to the circuit size. For a first empirical validation, we consider a simple application based on noisy seven-segment display images. The credal models are observed to properly distinguish between easy and hard-to-detect instances and outperform other generative models not able to cope with logical constraints.
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