Dirichlet Process Mixture of Generalized Inverted Dirichlet Distributions for Positive Vector Data With Extended...

Area of Science:

• Statistics
• Machine Learning
• Computational Statistics

Background:

• The generalized inverted Dirichlet distribution is effective for modeling positive-valued data vectors.
• Bayesian estimation of mixture models often faces computational challenges, particularly with variational inference (VI).
• Classical VI struggles with analytically calculating expectations for Dirichlet process (DP) mixtures of generalized inverted Dirichlet distributions (InGIDMM).

Purpose of the Study:

• To propose a novel Bayesian nonparametric approach for estimating InGIDMM.
• To address the computational limitations of classical VI in Bayesian estimation of InGIDMM.
• To develop a method that automatically determines the number of mixture components and avoids model underfitting/overfitting.

Main Methods:

• Utilizing a Dirichlet process (DP) mixture framework to create an infinite generalized inverted Dirichlet mixture model (InGIDMM).
• Implementing an extended variational inference (EVI) framework with lower bound approximations.
• Deriving analytically tractable solutions by overcoming the need for numerical simulations in posterior distribution estimation.

Main Results:

• The proposed extended VI (EVI) framework successfully overcomes the computational challenges of classical VI for InGIDMM.
• The DP mixture approach enables automatic determination of the number of mixture components from data.
• The InGIDMM effectively models positive-valued data vectors and avoids underfitting and overfitting issues.

Conclusions:

• The developed Bayesian nonparametric approach using EVI provides an efficient and analytically tractable method for InGIDMM estimation.
• This method offers a robust solution for modeling positive-valued data with automatic complexity selection.
• The approach demonstrates strong performance on both simulated and real-world datasets.