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[2602.12527] Analytical Results for Two Exponential Family Distributions in Hierarchical Dirichlet Processes
Summary
This paper explores analytical results for two exponential family distributions within the Hierarchical Dirichlet Process (HDP), focusing on the Poisson and normal distributions, and extends the applicability of HDP in Bayesian nonparametric modeling.
Why It Matters
The findings enhance the understanding of hierarchical Bayesian models by providing closed-form expressions for conjugate pairs, which can significantly improve the modeling of grouped data in various applications, making it relevant for researchers in machine learning and statistics.
Key Takeaways
- The paper derives explicit closed-form expressions for Gamma-Poisson and Normal-Gamma-Normal conjugate pairs.
- It extends the Hierarchical Dirichlet Process framework beyond traditional applications, enhancing its versatility.
- Detailed mathematical proofs clarify the structure of conjugacy in hierarchical nonparametric models.
- The results can aid researchers in employing hierarchical Bayesian nonparametrics more effectively.
- This work contributes to the broader understanding of exponential family distributions in machine learning.
Computer Science > Machine Learning arXiv:2602.12527 (cs) [Submitted on 13 Feb 2026] Title:Analytical Results for Two Exponential Family Distributions in Hierarchical Dirichlet Processes Authors:Naiqi Li View a PDF of the paper titled Analytical Results for Two Exponential Family Distributions in Hierarchical Dirichlet Processes, by Naiqi Li View PDF HTML (experimental) Abstract:The Hierarchical Dirichlet Process (HDP) provides a flexible Bayesian nonparametric framework for modeling grouped data with a shared yet unbounded collection of mixture components. While existing applications of the HDP predominantly focus on the Dirichlet-multinomial conjugate structure, the framework itself is considerably more general and, in principle, accommodates a broad class of conjugate prior-likelihood pairs. In particular, exponential family distributions offer a unified and analytically tractable modeling paradigm that encompasses many commonly used distributions. In this paper, we investigate analytic results for two important members of the exponential family within the HDP framework: the Poisson distribution and the normal distribution. We derive explicit closed-form expressions for the corresponding Gamma-Poisson and Normal-Gamma-Normal conjugate pairs under the hierarchical Dirichlet process construction. Detailed derivations and proofs are provided to clarify the underlying mathematical structure and to demonstrate how conjugacy can be systematically exploited in hierarchical non...