通过Hessian分数和三角运输学习非高斯图形模型
Learning non-Gaussian graphical models via Hessian scores and triangular transport
摘要：
无向概率图模型表示随机变量集合的条件依赖性或马尔可夫性。了解这种图形模型的稀疏性对于建模多变量分布和有效地执行推理是有价值的。虽然从数据中学习图结构的问题已经被广泛研究了某些参数分布族，但大多数现有的方法都无法一致地恢复非高斯数据的图结构。在这里，我们提出了一个学习连续和非高斯分布的马尔可夫结构的算法。为了表征条件独立性，我们引入了一个分数的基础上，从联合对数密度集成Hessian信息，我们证明了这个分数上限的条件互信息的一般类的分布。为了计算分数，我们的算法SING使用由三角运输图引起的确定性耦合来估计密度，并迭代地利用图中的稀疏结构来揭示图中的稀疏性。对于某些非高斯数据集，我们表明，我们的算法恢复的图形结构，即使有偏见的近似密度。在其他例子中，我们应用SING来学习具有局部相互作用的混沌动力系统的状态之间的依赖关系。
英文摘要：
Undirected probabilistic graphical models represent the conditional dependencies, or Markov properties, of a collection of random variables. Knowing the sparsity of such a graphical model is valuable for modeling multivariate distributions and for efficiently performing inference. While the problem of learning graph structure from data has been studied extensively for certain parametric families of distributions, most existing methods fail to consistently recover the graph structure for non-Gaussian data. Here we propose an algorithm for learning the Markov structure of continuous and non-Gaussian distributions. To characterize conditional independence, we introduce a score based on integrated Hessian information from the joint log-density, and we prove that this score upper bounds the conditional mutual information for a general class of distributions. To compute the score, our algorithm SING estimates the density using a deterministic coupling, induced by a triangular transport map, and iteratively exploits sparse structure in the map to reveal sparsity in the graph. For certain non-Gaussian datasets, we show that our algorithm recovers the graph structure even with a biased approximation to the density. Among other examples, we apply SING to learn the dependencies between the states of a chaotic dynamical system with local interactions.
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