Nonlinear systems of polynomial equations arise naturally in many applied settings, for example loglinear models on contingency tables and Gaussian graphical models. The solution sets to these systems over the reals are often positive dimensional spaces that in general may be very complicated yet have very nice local behavior almost everywhere. Standard methods in real algebraic geometry for describing positive dimensional real solution sets include cylindrical algebraic decomposition and numerical cell decomposition, both of which can be costly to compute in many practical applications. In this work we communicate recent progress towards a Monte Carlo framework for exploring such real solution sets. After describing how to construct probability distributions whose mass focuses on a variety of interest, we describe how Hamiltonian Monte Carlo methods can be used to sample points near the variety that may then be moved to the variety using endgames. We conclude by showcasing trial experiments using practical implementations of the method in the Bayesian engine Stan.

翻译：多项式方程组的非线性系统自然出现在许多应用场景中，例如列联表上的对数线性模型和高斯图模型。这些系统在实数域上的解集通常是正维空间，虽然整体结构可能非常复杂，但在几乎处处具有优良的局部性质。描述正维实解集的标准实代数几何方法包括柱形代数分解和数值胞腔分解，这两种方法在许多实际应用中计算代价可能很高。本文介绍了探索此类实解集的蒙特卡洛框架的最新进展。在阐述如何构建概率分布使其质量集中于目标簇后，我们说明了如何利用哈密顿蒙特卡洛方法采样簇附近的点，再通过终局算法将这些点移动至簇上。最后，我们通过贝叶斯引擎Stan中该方法的实际实现展示了初步实验效果。