Negative Binomial Process Count and Mixture Modeling (1209.3442v3)

Abstract: The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process for count modeling. A draw from the NB process consists of a Poisson distributed finite number of distinct atoms, each of which is associated with a logarithmic distributed number of data samples. We reveal relationships between various count- and mixture-modeling distributions and construct a Poisson-logarithmic bivariate distribution that connects the NB and Chinese restaurant table distributions. Fundamental properties of the models are developed, and we derive efficient Bayesian inference. It is shown that with augmentation and normalization, the NB process and gamma-NB process can be reduced to the Dirichlet process and hierarchical Dirichlet process, respectively. These relationships highlight theoretical, structural and computational advantages of the NB process. A variety of NB processes, including the beta-geometric, beta-NB, marked-beta-NB, marked-gamma-NB and zero-inflated-NB processes, with distinct sharing mechanisms, are also constructed. These models are applied to topic modeling, with connections made to existing algorithms under Poisson factor analysis. Example results show the importance of inferring both the NB dispersion and probability parameters.

• The paper proposes a unified model where the Negative Binomial process supports both count and mixture modeling through gamma-process rate measures and normalization techniques.
• It demonstrates efficient Bayesian inference via data augmentation, overcoming challenges in estimating overdispersion in count data.
• The study expands NB process variants such as beta-NB and zero-inflated-NB, broadening applications in topic modeling and handling sparse data.

Negative Binomial Process: Count and Mixture Modeling

This paper presents a comprehensive examination of the Negative Binomial (NB) process for count and mixture modeling, proposing innovative methods to integrate these two domains seamlessly. The authors unify count and mixture modeling by leveraging the relationships between Poisson, multinomial, gamma, and Dirichlet distributions. The paper particularly focuses on the NB process's capability to model both overdispersed count data and mixture components, essential in statistical and machine learning applications.

The NB process is established through a gamma process that acts as a rate measure for a Poisson process. This approach permits the derivation of a random probability measure suitable for mixture modeling, while its marginalization provides a framework for modeling count data through the NB process. The key insight lies in associating a finite number of distinct atoms—in Poisson distributed processes—with a logarithmically distributed quantity of data samples. The authors present how augmentation and normalization of the NB process, and the gamma-NB process can be efficiently converted into the Dirichlet process and the hierarchical Dirichlet process, respectively, showcasing substantial theoretical and computational benefits.

Key Findings

1. Unified Model and Augmentation Techniques:
   • The authors reveal that the NB process can effectively serve dual purposes: count modeling through the introduction of the gamma process, and mixture modeling by its normalized variant. The unifying nature of the Poisson-logarithmic bivariate distribution plays a significant role in connecting the NB process with traditional Chinese restaurant process distributions, thus elucidating deeper connections with nonparametric Bayesian approaches.
2. Efficient Bayesian Inference:
   • By applying data augmentation methods, the authors achieve analytic conditional posteriors enhancing Bayesian inference efficiency. Through insightful posterior analysis, the paper claims to solve some of the long-standing challenges associated with Bayesian inference for NB models, particularly around the dispersion parameters which are conventionally hard to infer accurately.
3. Expanding the NB Process Family:
   • The research extends the NB process to multiple variants including beta-NB and zero-inflated-NB processes. These processes provide flexibility in accommodating varied data structures, such as those exhibiting sparseness or overdispersion beyond the standard offerings of Dirichlet-like frameworks. The emphasis is laid on their application to topic modeling—showing the applicability to various problems such as Poisson factor analysis among others.
4. Potential Applications and Comparisons:
   • The paper benchmarks the NB process against hierarchical models like the HDP. Theoretical exploration shows that the NB process potentially addresses limitations found in Dirichlet process derivatives by offering a richer parameterization space, allowing improved model fit and interpretability, particularly in domains like document and topic modeling.

Implications and Future Directions

The implications of this work span both practical and theoretical domains in AI and statistics. Practically, the models proposed may enhance performance in fields requiring precise and flexible count data modeling, such as genomics and social sciences. Theoretically, the introduction of augmentation techniques and novel nonparametric processes opens avenues for further exploration of process and measure theory in statistical modeling.

Future work can explore three main avenues:

1. Extending the NB process to new data domains requiring flexible overdispersion handling features.
2. Deepening theoretical analysis of NB process variants to uncover additional computational efficiencies or model constraints.
3. Implementing and benchmarking these models in real-world applications beyond the scope of document modeling, such as in network traffic analysis or user behavior predictions.

In conclusion, the paper positions the NB process as a versatile tool capable of transcending traditional boundaries of count and mixture modeling, offering persuasive reasons for its adoption across different AI domains.