For exchangeable data, mixture models are an extremely useful tool for density estimation due to their attractive balance between smoothness and flexibility. When additional covariate information is present, mixture models can be extended for flexible regression by modeling the mixture parameters, namely the weights and atoms, as functions of the covariates. These types of models are interpretable and highly flexible, allowing non only the mean but the whole density of the response to change with the covariates, which is also known as density regression. This article reviews Bayesian covariate-dependent mixture models and highlights which data types can be accommodated by the different models along with the methodological and applied areas where they have been used. In addition to being highly flexible, these models are also numerous; we focus on nonparametric constructions and broadly organize them into three categories: 1) joint models of the responses and covariates, 2) conditional models with single-weights and covariate-dependent atoms, and 3) conditional models with covariate-dependent weights. The diversity and variety of the available models in the literature raises the question of how to choose among them for the application at hand. We attempt to shed light on this question through a careful analysis of the predictive equations for the conditional mean and density function as well as predictive comparisons in three simulated data examples.

翻译：对于可交换数据，混合模型因其在平滑性与灵活性之间取得的理想平衡，成为密度估计中极为有用的工具。当存在额外协变量信息时，可通过将混合参数（即权重和原子）建模为协变量的函数来扩展混合模型，从而实现灵活回归。此类模型兼具可解释性和高度灵活性，不仅允许响应变量的均值随协变量变化，还能实现其整个密度函数随协变量的变化（即密度回归）。本文综述了贝叶斯协变量依赖混合模型，重点阐明不同模型可处理的数据类型及其在方法论与应用领域的使用情况。除高度灵活性外，这类模型种类繁多；我们聚焦于非参数构造，并将其大致分为三类：（1）响应变量与协变量的联合模型，（2）具有单一权重和协变量依赖原子的条件模型，（3）具有协变量依赖权重的条件模型。文献中现有模型的多样性与差异性引出了如何针对具体应用选择模型的疑问。我们通过对条件均值与密度函数的预测方程进行细致分析，并结合三个模拟数据示例的预测比较，试图为这一问题提供见解。