伯努利随机变量和的树界:一种线性优化方法

Divya Padmanabhan, K. Natarajan
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引用次数: 3

摘要

我们研究了n个相关伯努利随机变量之和超过整数k的概率的最紧上界和下界的计算问题。在知道由完备图表示的所有双变量分布对的情况下,边界是NP难计算的。当在树图上指定二元分布时,我们证明了使用紧致线性程序在多项式时间内可以计算紧边界。当违反树结构图形模型中的条件独立性假设时,这些边界提供了鲁棒的概率估计。我们通过数值证明了我们的紧凑线性程序相对于其他方法的计算优势。提供了在各种知识假设(如单变量信息和条件独立性)下的边界的比较。在Chow–Liu树的上下文中说明了一个应用,其中我们的边界区分了编码最大可能互信息的各种树。
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Tree Bounds for Sums of Bernoulli Random Variables: A Linear Optimization Approach
We study the problem of computing the tightest upper and lower bounds on the probability that the sum of n dependent Bernoulli random variables exceeds an integer k. Under knowledge of all pairs of bivariate distributions denoted by a complete graph, the bounds are NP-hard to compute. When the bivariate distributions are specified on a tree graph, we show that tight bounds are computable in polynomial time using a compact linear program. These bounds provide robust probability estimates when the assumption of conditional independence in a tree-structured graphical model is violated. We demonstrate, through numericals, the computational advantage of our compact linear program over alternate approaches. A comparison of bounds under various knowledge assumptions, such as univariate information and conditional independence, is provided. An application is illustrated in the context of Chow–Liu trees, wherein our bounds distinguish between various trees that encode the maximum possible mutual information.
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