Bayesian inference for Dirichlet-Multinomial (DM) models has a long and important history. The concentration parameter $\alpha$ is pivotal in smoothing category probabilities within the multinomial distribution and is crucial for the inference afterward. Due to the lack of a tractable form of its marginal likelihood, $\alpha$ is often chosen in an ad-hoc manner, or estimated using approximation algorithms. A constant $\alpha$ often leads to inadequate smoothing of probabilities, particularly for sparse compositional count datasets. In this paper, we introduce a novel class of prior distributions facilitating conjugate updating of the concentration parameter, allowing for full Bayesian inference for DM models. Our methodology is based on fast residue computation and admits closed-form posterior moments in specific scenarios. Additionally, our prior provides continuous shrinkage with its heavy tail and substantial mass around zero, ensuring adaptability to the sparsity or quasi-sparsity of the data. We demonstrate the usefulness of our approach on both simulated examples and on real-world applications. Finally, we conclude with directions for future research.

翻译：狄利克雷-多项（DM）模型的贝叶斯推断具有悠久而重要的历史。浓度参数α在平滑多项分布中的类别概率方面起着关键作用，并对后续的推断至关重要。由于边际似然缺乏易处理的形式，α通常以临时方式选择，或使用近似算法进行估计。常数α往往导致概率平滑不足，尤其是在稀疏成分计数数据集中。本文引入了一类新颖的先验分布，促进浓度参数的共轭更新，允许对DM模型进行全贝叶斯推断。我们的方法基于快速残差计算，并在特定场景下提供闭式后验矩。此外，我们的先验通过其重尾和围绕零的大量质量提供连续收缩，确保对数据稀疏性或准稀疏性的适应性。我们在模拟示例和实际应用中展示了该方法的有效性。最后，我们总结了未来研究的方向。