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：混合成员模型（或称部分成员模型）是一种灵活的無监督学习方法，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成员模型。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，该表示在保持协方差结构数据驱动学习的同时实现高效计算。在该框架下，我们证明了在已知特征分配矩阵条件下的条件后验一致性。与以往混合成员模型研究相比，本方法在增加建模灵活性的同时，可直接解释均值与协方差结构。本研究受基于脑电图（EEG）对自闭症谱系障碍（ASD）儿童进行功能性脑成像研究的驱动。在此背景下，本文通过特征隶属度比例正式形式化了临床上"谱系"的概念。