标题： 基于依赖标签的高斯混合模型推断 显示英文标题

标题： Inference on Gaussian mixture models with dependent labels

摘要： 高斯混合模型被广泛用于建模来自多个潜在源的数据。 尽管其受欢迎，大多数理论研究假设标签是独立同分布的，或者遵循马尔可夫链。 在更复杂的依赖关系下，估计的基本极限如何变化仍不清楚。 在本文中，我们针对球面两组分高斯混合模型解决了这个问题。 我们首先证明，对于具有任意依赖关系的标签，基于错误指定似然的朴素估计量是$\sqrt{n}$-一致的。 此外，在标签遵循伊辛模型的情况下，我们建立了估计的信息论限制，并发现随着依赖性的增强出现了一个有趣的相变。 当依赖性小于阈值时，最优估计量及其极限方差恰好与独立情况相同，适用于一大类伊辛模型。 另一方面，在更强的依赖性下，估计变得更容易，朴素估计量不再是最佳的。 因此，我们提出了一种基于似然函数变分近似的替代估计量，并在特定的伊辛模型下论证了其最优性。 显示英文摘要

摘要： Gaussian mixture models are widely used to model data generated from multiple latent sources. Despite its popularity, most theoretical research assumes that the labels are either independent and identically distributed, or follows a Markov chain. It remains unclear how the fundamental limits of estimation change under more complex dependence. In this paper, we address this question for the spherical two-component Gaussian mixture model. We first show that for labels with an arbitrary dependence, a naive estimator based on the misspecified likelihood is $\sqrt{n}$-consistent. Additionally, under labels that follow an Ising model, we establish the information theoretic limitations for estimation, and discover an interesting phase transition as dependence becomes stronger. When the dependence is smaller than a threshold, the optimal estimator and its limiting variance exactly matches the independent case, for a wide class of Ising models. On the other hand, under stronger dependence, estimation becomes easier and the naive estimator is no longer optimal. Hence, we propose an alternative estimator based on the variational approximation of the likelihood, and argue its optimality under a specific Ising model.