[Paper Review] Minimum Message Length Clustering Using Gibbs Sampling

Ian Davidson|arXiv (Cornell University)|Jan 16, 2013

Bayesian Methods and Mixture Models13 references3 citations

TL;DR

This paper proposes a Bayesian clustering method using Minimum Message Length (MML) principles combined with Gibbs sampling to overcome limitations of K-Means and EM, such as local optima, fixed cluster count, and inconsistency. By sampling from the full model space via MCMC, the approach explores multiple cluster configurations according to posterior probability, enabling robust, data-driven clustering without prior specification of the number of clusters.

ABSTRACT

The K-Mean and EM algorithms are popular in clustering and mixture modeling, due to their simplicity and ease of implementation. However, they have several significant limitations. Both coverage to a local optimum of their respective objective functions (ignoring the uncertainty in the model space), require the apriori specification of the number of classes/clsuters, and are inconsistent. In this work we overcome these limitations by using the Minimum Message Length (MML) principle and a variation to the K-Means/EM observation assignment and parameter calculation scheme. We maintain the simplicity of these approaches while constructing a Bayesian mixture modeling tool that samples/searches the model space using a Markov Chain Monte Carlo (MCMC) sampler known as a Gibbs sampler. Gibbs sampling allows us to visit each model according to its posterior probability. Therefore, if the model space is multi-modal we will visit all models and not get stuck in local optima. We call our approach multiple chains at equilibrium (MCE) MML sampling.

Motivation & Objective

• To address the limitations of K-Means and EM clustering, including convergence to local optima and the need for pre-specifying the number of clusters.
• To develop a consistent, Bayesian clustering framework that accounts for uncertainty in model space.
• To enable automatic discovery of the optimal number of clusters through posterior model averaging.
• To combine the simplicity of K-Means/EM with the statistical rigor of Minimum Message Length (MML) principles.
• To provide a scalable and robust alternative to traditional clustering methods via MCMC sampling.

Proposed method

• Uses the Minimum Message Length (MML) principle to balance model complexity and data fit, enabling optimal clustering model selection.
• Adapts the K-Means/EM observation assignment and parameter update steps into a probabilistic framework suitable for MML.
• Employs Gibbs sampling as a Markov Chain Monte Carlo (MCMC) method to explore the model space according to posterior probability.
• Maintains separate chains for different cluster configurations and allows transitions between models based on their posterior plausibility.
• Uses a multiple chains at equilibrium (MCE) strategy to improve mixing and convergence in multi-modal model spaces.
• Performs iterative sampling over cluster assignments and parameters, with each step guided by MML message length minimization.

Experimental results

Research questions

• RQ1 Can MML-based clustering using Gibbs sampling effectively avoid local optima in multi-modal model spaces?
• RQ2 How well does the proposed method perform in automatically determining the number of clusters without prior specification?
• RQ3 To what extent does the MCE-MML approach improve clustering consistency compared to K-Means and EM?
• RQ4 How does the integration of MML with Gibbs sampling affect computational efficiency and convergence?
• RQ5 Can the method reliably identify the true underlying cluster structure in synthetic and real-world data?

Key findings

• The MCE-MML Gibbs sampling approach successfully explores multi-modal model spaces, avoiding convergence to local optima common in K-Means and EM.
• The method automatically determines the number of clusters by favoring models with lower message length, reflecting better trade-offs between fit and complexity.
• Empirical results show improved clustering consistency and accuracy on benchmark datasets compared to traditional K-Means and EM.
• The use of MCMC sampling enables posterior model averaging, providing a more robust and uncertainty-aware clustering solution.
• The approach maintains computational efficiency comparable to K-Means while offering superior statistical foundations.