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.

翻译：高斯混合模型广泛用于建模来自多个潜在源生成的数据。尽管其应用广泛，但大多数理论研究假设标签要么独立同分布，要么遵循马尔可夫链。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。本文针对球形双分量高斯混合模型探讨了这一问题。我们首先证明，对于具有任意依赖关系的标签，基于误设似然的朴素估计量具有$\sqrt{n}$一致性。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理论极限，并发现了随着依赖性增强而出现的相变现象。当依赖性低于阈值时，对于一大类伊辛模型，最优估计量及其极限方差与独立情况完全一致。另一方面，在更强依赖性下，估计变得更容易且朴素估计量不再最优。因此，我们提出了一种基于似然变分近似的替代估计量，并论证了其在特定伊辛模型下的最优性。