Probabilistic graphical models are widely used to model complex systems with uncertainty. Traditionally, Gaussian directed graphical models are applied for analysis of large networks with continuous variables since they can provide conditional and marginal distributions in closed form simplifying the inferential task. The Gaussianity and linearity assumptions are often adequate, yet can lead to poor performance when dealing with some practical applications. In this paper, we model each variable in graph G as a polynomial regression of its parents to capture complex relationships between individual variables and with utility function of polynomial form. Since the marginal posterior distributions of individual variables can become analytically intractable, we develop a message-passing algorithm to propagate information throughout the network solely using moments which enables the expected utility scores to be calculated exactly. We illustrate how the proposed methodology works in a decision problem in energy systems planning.

翻译：概率图模型被广泛用于对具有不确定性的复杂系统进行建模。传统上，高斯有向图模型因其能够以封闭形式提供条件分布和边缘分布，从而简化推理任务，而被应用于分析具有连续变量的大型网络。尽管高斯性和线性假设通常足够，但在处理某些实际应用时，这些假设可能导致性能不佳。本文中，我们将图G中的每个变量建模为其父变量的多项式回归，以捕捉单个变量之间的复杂关系，并采用多项式形式的效用函数。由于单个变量的边缘后验分布可能难以解析求解，我们开发了一种消息传递算法，仅利用矩在网络中传播信息，从而能够精确计算期望效用得分。我们通过能源系统规划中的一个决策问题，展示了所提方法的应用效果。