Combinatorial Clustering and the Beta Negative Binomial Process.

Area of Science:

• Statistics
• Machine Learning
• Computational Statistics

Background:

• Traditional latent class models often assume exclusivity and single class membership.
• Handling overlapping and repeated class memberships requires more flexible modeling approaches.

Purpose of the Study:

• To develop a Bayesian nonparametric approach for general latent class problems with simultaneous and multiple class memberships.
• To introduce the negative binomial process (NBP) as a suitable prior for these complex scenarios.

Main Methods:

• Introduced the negative binomial process (NBP) as an infinite-dimensional prior.
• Established conjugacy between NBP and the beta process, defining the beta-negative binomial process (BNBP).
• Developed hierarchical models (HBNBP) and derived Markov Chain Monte Carlo (MCMC) algorithms for posterior inference.

Main Results:

• Characterized posterior distributions under BNBP and HBNBP.
• Studied asymptotic properties of BNBP and proposed a three-parameter extension with power-law behavior.
• Demonstrated the utility of HBNBP through MCMC algorithms in image segmentation, object recognition, and document analysis.

Conclusions:

• The developed Bayesian nonparametric framework, particularly the HBNBP, provides a powerful tool for analyzing complex latent class structures.
• The NBP and its extensions offer significant flexibility for modeling data with overlapping and repeated class memberships across various domains.