dirichlet-multinomial modeling

This study addresses estimation bias and hypothesis test failure arising from frequent zero counts in categorical data analysis. Methodologically: (1) it develops a robust Bayesian estimator for the binomial distribution; (2) it designs a regularized maximum likelihood–based hypothesis testing framework, covering sign tests, homogeneity tests, and symmetry tests; and (3) it introduces analytically tractable regularized measures of association for contingency tables—including regularized mutual information—that explicitly accommodate zero frequencies. The contributions are threefold: first, substantially improved stability and interpretability of probability estimation, hypothesis testing, and association assessment under data sparsity; second, provision of a theoretically rigorous yet computationally feasible foundational toolkit for high-dimensional sparse categorical data; and third, enhanced reliability of practical statistical inference in real-world applications involving sparse contingency structures.

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 nonparametric models. Our work extends the applicability of the HDP beyond the Dirichlet-multinomial setting and furnishes practical analytic results for researchers employing hierarchical Bayesian nonparametrics.

This study addresses the challenge of specifying concentration parameter priors in Dirichlet process mixture models, where default hyperpriors often impose overly strong or unintended assumptions. To resolve this, the authors propose the Design Conditional prior Elicitation (DCE) framework, which translates practitioners’ prior beliefs about clustering structure into a Gamma hyperprior tailored to a fixed sample size, thereby jointly regulating the number of clusters and the distribution of mixture weights. The approach employs a two-stage moment-matching procedure to enhance computational efficiency and introduces a dual-anchor protocol to diagnose and mitigate risks of weight dominance. Experiments demonstrate that DCE-calibrated priors substantially reduce posterior collapse rates—by over 60% compared to the default Gamma(1,1)—and consistently improve clustering accuracy and robustness across varying data informativeness. An open-source R package, DPprior, and reproducible diagnostic workflows are provided.

Bayesian multiple testing correction for simultaneous comparisons among $n$ groups faces combinatorial explosion: the number of possible equality-constrained partitions equals the Bell number $ ext{Bell}(n)$—e.g., 115,975 for $n=10$—rendering exhaustive enumeration infeasible. Method: We propose a scalable Bayesian framework using a Beta-binomial prior to model equality structures as group partitions, integrated with a Dirichlet process prior and a stochastic search algorithm for efficient posterior exploration. Contribution/Results: Our approach enables, for the first time, joint inference on equality constraints over means, standard deviations, and proportions across the full partition space. We implement this methodology in EqualitySampler—a high-performance Julia package supporting large-scale, high-precision identification of equality structures. In both simulations and empirical applications, EqualitySampler significantly improves accuracy and interpretability of model selection under multiple comparisons.

Conventional statistical software frequently encounters infeasibility issues in estimating cumulative link model (CLM) parameters, and standard CLMs lack flexibility in handling missing responses, longitudinal binary outcomes, and non-proportional odds structures. Method: We propose a novel family of regression models for ordinal responses—comprising mixed-link, two-group, conditional-link, and PO–NPO hybrid specifications—that rigorously characterize the feasible parameter space of CLMs for the first time, providing necessary and sufficient feasibility conditions. We develop a verifiable maximum likelihood estimation (MLE) feasibility algorithm, derive closed-form expressions for the Fisher information matrix, and construct a comprehensive model selection framework incorporating AIC and BIC. Contributions/Results: Our approach relaxes the proportional odds assumption, enabling category-specific modeling. Empirical results demonstrate substantially improved goodness-of-fit, correction of misclassification induced by missing responses (NA), resolution of CLM convergence failures in mainstream software, and more robust and accurate statistical inference.

本文提出了一种非参数贝叶斯推理框架，用于解决部分识别的离散响应模型问题，通过直接对条件概率质量函数进行推理，避免了将条件矩转换为无条件矩或离散化协变量的需求。

This study addresses the inefficiency of Markov chain Monte Carlo (MCMC) sampling in Bayesian models for highly zero-inflated count data, where strong posterior dependencies hinder effective exploration. The work proposes the first integration of marginal data augmentation into a semiparametric Bayesian count regression framework. By introducing working parameters to rescale latent variables associated with zero observations, the method substantially alleviates posterior correlations. This approach markedly improves MCMC mixing and convergence rates, outperforming existing sampling strategies in both synthetic experiments and real-world modeling of subnational mortality counts in Austria. The proposed technique thus offers an efficient and reliable Bayesian inference pathway for high-dimensional zero-inflated count data.

This work addresses the challenge of modeling multivariate count data with exclusion or incompatibility constraints on graph-structured variables by proposing a unified graphical distribution framework. It systematically constructs, for the first time, four families of distributions—graphical multinomial, negative multinomial, hypergeometric, and negative hypergeometric—leveraging decomposable graphs to encode variable dependencies and feasible configurations. The framework supports continuous interpolation between the empty and complete graphs while preserving an explicit Markov factorization and tractable sampling schemes. Building upon graphical Dirichlet-type priors, a Bayesian hierarchical model is developed, yielding closed-form posterior and predictive distributions. The approach demonstrates both flexibility and practical utility in constrained counting tasks, such as Rydberg atom excitation experiments.

This work addresses the challenge of Bayesian inference in non-conjugate distance-dependent Chinese Restaurant Processes (ddCRP), where the lack of conjugacy between cluster parameters and the likelihood leads to a parameter space of varying dimensionality. To tackle this, the authors develop a reversible-jump Markov chain Monte Carlo (RJMCMC) framework that incorporates multiple birth-death move strategies based on prior matching, independence, and moment matching with respect to the data. A posterior resampling mechanism is introduced to enhance the acceptance rate of fixed-dimensional moves. By combining moment-matching proposal distributions with resampling techniques, the proposed method accommodates both discrete and continuous observation models. Experiments on synthetic data and the Old Faithful geyser eruption dataset demonstrate that moment-matching proposals substantially outperform conventional prior-based proposals, yielding efficient and accurate inference for non-conjugate ddCRP models.

This study addresses the overdispersion problem in finite discrete count data and the limitations of the Beta-binomial model, including its subjective prior specification and fitting bias. To overcome these issues, this work proposes the Bernstein-binomial model. By leveraging the uniform approximation property of Bernstein polynomials, the method constructs a highly flexible prior framework that transcends traditional unimodal constraints, dynamically adapts to complex data structures, and supports both frequentist and Bayesian inference as well as regression extensions. Simulation experiments and empirical analyses demonstrate that the proposed model significantly outperforms existing methods in terms of fitting accuracy, robustness, and predictive performance.