  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.
