A Novel Class of Pochhammer Priors for Bayesian Inference in Sparse Count Models, with Applications to Dirichlet-Multinomial Models

Core Concepts

This paper introduces a novel class of Pochhammer priors for Bayesian inference in count models, addressing the challenges of concentration parameter inference and handling sparse or quasi-sparse count data, particularly in Dirichlet-Multinomial models.

Abstract

• Bibliographic Information: Wang, Y., & Polson, N. G. (2024). Pochhammer Priors for Sparse Count Models. arXiv preprint arXiv:2402.09583v3.
• Research Objective: This paper proposes a new class of prior distributions, called Pochhammer priors, to address two key challenges in Bayesian inference for count models: (1) Difficulty in inferring the concentration parameter in the presence of Gamma function ratios. (2) Inability to effectively handle sparse or quasi-sparse count data.
• Methodology: The authors leverage the properties of Pochhammer polynomials and partial fraction decomposition to construct conjugate priors for the concentration parameter in Dirichlet-Multinomial (DM) models. They introduce two classes of priors: Pochhammer and Power Pochhammer distributions. For heterogeneous DM models, a Metropolis-Within-Gibbs sampling scheme is employed for posterior inference. The authors recommend a default "Half-Horseshoe" prior configuration (m=0, b=2) within the Pochhammer family to induce continuous shrinkage and adapt to sparsity.
• Key Findings: The proposed Pochhammer priors enable closed-form posterior moments for homogeneous DM models and facilitate efficient marginal posterior inference for the concentration parameter. The Half-Horseshoe prior, in particular, exhibits desirable properties for sparse data analysis, including substantial mass around zero and a heavy tail, leading to automatic adaptation to sparsity or quasi-sparsity. The authors demonstrate the efficacy of their approach through simulations and real-world applications.
• Main Conclusions: The Pochhammer prior family offers a powerful tool for Bayesian inference in count models, particularly those involving Gamma function ratios. The proposed priors provide computational advantages, including conjugate updating and closed-form posterior moments, while effectively handling sparse count data. The authors suggest potential generalizations of their approach to other count models, such as Negative Binomial and Generalized Dirichlet-Multinomial distributions.
• Significance: This research contributes significantly to Bayesian statistics and machine learning by providing a novel and efficient method for inference in count models, which are widely used in various domains. The ability to handle sparsity effectively is particularly relevant for modern high-dimensional datasets.
• Limitations and Future Research: The authors acknowledge the computational challenges associated with high-dimensional heterogeneous DM models and suggest exploring direct sampling methods for the Pochhammer Gibbs conditional distribution in future work. Further research could investigate the application of Pochhammer priors to a wider range of count models and explore their theoretical properties in greater depth.

Stats

K = 100 (number of categories)
N = 50 (total counts)

Quotes

Key Insights Distilled From

Pochhammer Priors for Sparse Count Models

by Yuexi Wang, ... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2402.09583.pdf

Deeper Inquiries

How do Pochhammer priors compare to other sparsity-inducing priors, such as the horseshoe prior or the spike-and-slab prior, in terms of computational efficiency and statistical performance for count data?

Pochhammer priors, particularly the "half-horseshoe" variant, offer a unique blend of computational efficiency and strong statistical performance for sparse count data when compared to traditional sparsity-inducing priors like horseshoe or spike-and-slab priors. Here's a breakdown:
Computational Efficiency:

Pochhammer (Half-Horseshoe): These priors shine due to their conjugacy with the Dirichlet-Multinomial model. This leads to closed-form posterior updates for the concentration parameter, significantly reducing computational burden, especially in high dimensions.
Horseshoe: While effective at inducing sparsity, horseshoe priors lack conjugacy with count models. Inference typically relies on computationally intensive MCMC methods, which can become slow in high-dimensional settings.
Spike-and-Slab: These priors are also computationally demanding. They involve a mixture of a point mass at zero (spike) and a continuous distribution (slab), requiring either MCMC or variational approximations for inference.
Statistical Performance:

Pochhammer (Half-Horseshoe): Offers continuous shrinkage, effectively handling both true zeros and near-zero values in count data. This adaptability makes it suitable for a wider range of sparsity patterns compared to the strict zero-inflation approach.
Horseshoe: Exhibits similar continuous shrinkage properties, performing well in recovering sparse signals. However, the lack of conjugacy with count models can lead to more complex and potentially less efficient inference.
Spike-and-Slab: Provides strong sparsity by explicitly setting some coefficients to zero. However, the discrete nature of the spike can be less flexible for count data, which often exhibit near-zero values rather than perfect zeros.
In summary: Pochhammer priors, specifically the half-horseshoe variant, stand out for sparse count data analysis. Their conjugacy with Dirichlet-Multinomial models enables faster computation, while their continuous shrinkage properties allow for flexible adaptation to various sparsity patterns.

Could the proposed Pochhammer priors be extended to handle count data with negative correlations, potentially by incorporating them into models like the Negative Multinomial distribution?

Yes, the paper demonstrates that Pochhammer priors can be extended to handle count data with more complex correlation structures, including negative correlations, by incorporating them into models beyond the Dirichlet-Multinomial, such as the Negative Binomial and Generalized Dirichlet-Multinomial distributions.
Here's how:

Negative Binomial: The paper shows that by using a coupled prior structure where the probability parameter (π) of the Negative Binomial distribution is dependent on the dispersion parameter (α) through a Beta distribution, and placing a Pochhammer prior on α, one can achieve conjugate updates and closed-form posterior moments. This approach allows for flexible modeling of overdispersed count data, which often exhibit negative correlations.

Generalized Dirichlet-Multinomial (GDM): The GDM model allows for both positive and negative correlations between counts. The paper proposes using a half-horseshoe Pochhammer prior on the α parameters of the Beta distributions in the stick-breaking construction of the GDM. This encourages shrinkage towards zero for some probabilities, effectively capturing sparsity patterns while accounting for the underlying correlation structure.
Therefore, the flexibility of Pochhammer priors in conjunction with their computational advantages makes them suitable for a wider range of count data models, including those that capture negative correlations.

What are the potential implications of using Pochhammer priors in real-world applications where interpretability of the model is crucial, such as topic modeling or microbiome data analysis?

Using Pochhammer priors, especially the half-horseshoe variant, in real-world applications like topic modeling or microbiome data analysis, where interpretability is key, presents both opportunities and challenges:
Potential Advantages:

Enhanced Sparsity and Interpretability: The shrinkage properties of Pochhammer priors can lead to sparser models. In topic modeling, this translates to topics with fewer but more relevant words, making them easier to interpret. Similarly, in microbiome analysis, it can highlight key microbial taxa associated with specific conditions.
Improved Model Fit: By accommodating both zero and near-zero values, Pochhammer priors can lead to better model fit, especially in datasets with varying sparsity levels. This is crucial in microbiome data, where many taxa are rare and their abundances are often low.
Computational Efficiency: The conjugacy of Pochhammer priors with certain count models can significantly speed up computation, making them suitable for large datasets often encountered in these applications.
Potential Challenges:

Prior Sensitivity: While the paper suggests default hyperparameters for the Pochhammer prior, the impact of these choices on the resulting sparsity and interpretation needs careful consideration. Sensitivity analysis and prior elicitation techniques might be necessary.
Complex Interpretation of Shrinkage: Unlike spike-and-slab priors, which explicitly set some coefficients to zero, the continuous shrinkage induced by Pochhammer priors can be harder to interpret directly. Understanding the degree and pattern of shrinkage is crucial for drawing meaningful conclusions.
Overall: Pochhammer priors offer a promising avenue for enhancing interpretability in applications like topic modeling and microbiome analysis by promoting sparsity and improving model fit. However, careful consideration of prior sensitivity and the interpretation of shrinkage patterns is essential for drawing meaningful and reliable insights from the data.