The hierarchical Dirichlet process is the cornerstone of Bayesian nonparametric multilevel models. Its generative model can be described through a set of latent variables, commonly referred to as tables within the popular restaurant franchise metaphor. The latent tables simplify the expression of the posterior and allow for the implementation of Gibbs sampling algorithms to approximately draw posterior samples. However, managing their assignments can become computationally expensive, especially as the size of the dataset and the number of levels increase. In this work, we identify a prior for the concentration parameter of the hierarchical Dirichlet process that (i) induces a quasi-conjugate posterior distribution, and (ii) removes the need for tables, leading to more interpretable expressions for the posterior, with both a scalable and an exact algorithm to sample from it. Remarkably, this construction extends beyond the Dirichlet process, leading to a new framework for defining normalized hierarchical random measures and a new class of algorithms to sample from their posteriors. The key analytical tool is the independence of multivariate increments, that is, their representation as completely random vectors.

翻译：层次狄利克雷过程是贝叶斯非参数多层级模型的基石。其生成模型可通过一组潜变量来描述，这些潜变量在流行的餐馆特许经营隐喻中通常被称为“表”。潜变量表简化了后验表达，并允许实现吉布斯采样算法来近似抽取后验样本。然而，随着数据集规模和层级数量的增加，管理这些表的分配在计算上可能变得昂贵。在本文中，我们为层次狄利克雷过程的浓度参数确定了一种先验分布，该先验（i）诱导出拟共轭的后验分布，且（ii）消除了对表的需求，从而得到更具可解释性的后验表达式，同时提供了可伸缩和精确的采样算法。值得注意的是，此构造超越了狄利克雷过程，为定义归一化层次随机测度提供了一种新框架，并开创了一类从其后验进行采样的新算法。关键分析工具是多元增量的独立性，即将其表示为完全随机向量。