Mixed membership models, or partial membership models, are a flexible unsupervised learning method that allows each observation to belong to multiple clusters. In this paper, we propose a Bayesian mixed membership model for functional data. By using the multivariate Karhunen-Lo\`eve theorem, we are able to derive a scalable representation of Gaussian processes that maintains data-driven learning of the covariance structure. Within this framework, we establish conditional posterior consistency given a known feature allocation matrix. Compared to previous work on mixed membership models, our proposal allows for increased modeling flexibility, with the benefit of a directly interpretable mean and covariance structure. Our work is motivated by studies in functional brain imaging through electroencephalography (EEG) of children with autism spectrum disorder (ASD). In this context, our work formalizes the clinical notion of "spectrum" in terms of feature membership proportions.

翻译：混合隶属度模型（或称部分隶属度模型）是一种灵活的元监督学习方法，允许每个观测值同时归属于多个聚类。本文针对函数型数据提出一种贝叶斯混合隶属度模型。通过利用多元Karhunen-Loève定理，我们推导出高斯过程的可扩展表示，该表示能够保持数据驱动的协方差结构学习。在此框架下，我们建立了给定已知特征分配矩阵的条件后验一致性。相较于既有混合隶属度模型研究，本文所提方法在提升建模灵活性的同时，保留了可直接解释的均值与协方差结构。本研究受自闭症谱系障碍（ASD）儿童脑电图（EEG）功能成像研究的启发，在该应用背景下，我们以特征隶属度比例形式正式化了临床"谱系"概念。