We develop a modeling framework for joint factor and cluster analysis of datasets where multiple categorical response items are collected on a heterogeneous population of individuals. We introduce a latent factor multinomial probit model and employ prior constructions that allow inference on the number of factors as well as clustering of the subjects into homogenous groups according to their relevant factors. Clustering, in particular, allows us to borrow strength across subjects, therefore helping in the estimation of the model parameters, particularly when the number of observations is small. We employ Markov chain Monte Carlo techniques and obtain tractable posterior inference for our objectives, including sampling of missing data. We demonstrate the effectiveness of our method on simulated data. We also analyze two real-world educational datasets and show that our method outperforms state-of-the-art methods. In the analysis of the real-world data, we uncover hidden relationships between the questions and the underlying educational concepts, while simultaneously partitioning the students into groups of similar educational mastery.

中文翻译：

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用于分析多个分类项目响应的贝叶斯非参数方法

我们为数据集的联合因子和聚类分析开发了一个建模框架，其中在异质人群中收集了多个分类响应项。我们引入了一个潜在因子多项概率模型，并采用了先验结构，允许推断因子的数量以及根据相关因子将受试者聚类到同质组中。特别是聚类，允许我们借用跨学科的力量，因此有助于估计模型参数，特别是当观察数量很少时。我们采用马尔可夫链蒙特卡罗技术，为我们的目标获得易于处理的后验推理，包括对缺失数据的采样。我们证明了我们的方法对模拟数据的有效性。我们还分析了两个真实世界的教育数据集，并表明我们的方法优于最先进的方法。在对真实世界数据的分析中，我们揭示了问题与潜在教育概念之间的隐藏关系，同时将学生划分为具有相似教育水平的组。