概率括号符号：多变量系统和静态贝叶斯网络
Probability Bracket Notation: Multivariable Systems and Static Bayesian Networks
AI总结：
本文扩展了概率括号符号（PBN）以适用于多变量概率系统和静态贝叶斯网络，通过统一的代数形式定义概率分布和条件期望，提供系统的方法来表示和操作随机变量间的依赖关系，并展示了其在推理和期望计算中的应用。
摘要：
我们扩展了概率括号符号（PBN），受量子力学Dirac符号的启发，应用于多变量概率系统和静态贝叶斯网络（BNs）。通过定义概率分布和条件期望的统一、基无关的代数形式，PBN提供了一种系统的方法来表示和操作随机变量间的依赖关系。使用著名的Student BN作为示例概率图模型，我们展示了在PBN框架内的预测、自下而上和自上而下的推断以及期望计算。我们证明，对于具有N个节点的二元BN，在一次预处理后，沿着具有k个中间节点的d分离链进行推断需要O(k2^k)次操作，与直接从联合分布计算所需的O(N2^N)次操作相比显著减少。我们进一步将PBN扩展到具有连续变量的网络，包括线性高斯模型，并引入了一个混合医疗BN，结合了离散和连续变量。在该模型中，离散显示节点用作连续父节点的代理，使用户特定的预测成为可能。总体而言，PBN提供了一个基于操作的框架，统一了表示和计算，潜在应用包括教育、数据分析和机器学习。
英文摘要：
We extend Probability Bracket Notation (PBN), inspired by the Dirac notation in quantum mechanics, to multivariable probability systems and static Bayesian networks (BNs). By defining probability distributions and conditional expectations in a unified, basis-independent algebraic form, PBN provides a systematic way to represent and manipulate dependencies among random variables. Using the well-known Student BN as an illustrative probabilistic graphical model, we demonstrate prediction, bottom-up and top-down inference, and expectation calculations within the PBN framework. We show that, for a large N-node binary BN, after a one-time preprocessing, inference along a d-separable chain with k intermediate nodes requires O(k2^k) operations, compared to O(N2^N) for direct computation from the full joint distribution. We further extend PBN to networks with continuous variables, including linear Gaussian models, and introduce a hybrid Healthcare BN that combines discrete and continuous variables. In this model, discrete-display nodes serve as proxies for continuous parents, enabling user-specific predictions. Overall, PBN provides an operator-based framework that unifies representation and computation, with potential applications in education, data analytics, and machine learning.
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