Covariance matrices arise naturally in different scientific fields, including finance, genomics, and neuroscience, where they encode dependence structures and reveal essential features of complex multivariate systems. In this work, we introduce a comprehensive Bayesian framework for analyzing heterogeneous covariance data through both classical mixture models and a novel mixture-of-experts Wishart (MoE-Wishart) model. The proposed MoE-Wishart model extends standard Wishart mixtures by allowing mixture weights to depend on predictors through a multinomial logistic gating network. This formulation enables the model to capture complex, nonlinear heterogeneity in covariance structures and to adapt subpopulation membership probabilities to covariate-dependent patterns. To perform inference, we develop an efficient Gibbs-within-Metropolis-Hastings sampling algorithm tailored to the geometry of the Wishart likelihood and the gating network. We additionally derive an Expectation-Maximization algorithm for maximum likelihood estimation in the mixture-of-experts setting. Extensive simulation studies demonstrate that the proposed Bayesian and maximum likelihood estimators achieve accurate subpopulation recovery and estimation under a range of heterogeneous covariance scenarios. Finally, we present an innovative application of our methodology to a challenging dataset: cancer drug sensitivity profiles, illustrating the ability of the MoE-Wishart model to leverage covariance across drug dosages and replicate measurements. Our methods are implemented in the \texttt{R} package \texttt{moewishart} available at https://github.com/zhizuio/moewishart .

翻译：协方差矩阵在金融学、基因组学和神经科学等不同科学领域中自然产生，它们编码依赖结构并揭示复杂多元系统的本质特征。本文提出一个全面的贝叶斯框架，通过经典混合模型及新型的专家混合Wishart（MoE-Wishart）模型来分析异质性协方差数据。所提出的MoE-Wishart模型通过让混合权重通过多项式逻辑门控网络依赖于预测变量，扩展了标准Wishart混合模型。该公式使模型能够捕捉协方差结构中复杂的非线性异质性，并使亚群隶属概率适应协变量依赖模式。为进行推断，我们针对Wishart似然函数和门控网络的几何特性，开发了一种高效的Gibbs-within-Metropolis-Hastings抽样算法。此外，我们推导了专家混合设定下最大似然估计的期望最大化算法。大量模拟研究表明，所提出的贝叶斯估计量和最大似然估计量在多种异质性协方差场景下均能实现准确的亚群恢复和估计。最后，我们将该方法创新性地应用于一个具有挑战性的数据集：癌症药物敏感性谱，展示了MoE-Wishart模型利用跨药物剂量和重复测量的协方差的能力。我们的方法已实现于\texttt{R}软件包\texttt{moewishart}中，可通过https://github.com/zhizuio/moewishart 获取。