A Powerful Bayesian Test for Equality of Means in High Dimensions

Area of Science:

• Statistics
• Bayesian inference
• High-dimensional data analysis

Background:

• Comparing population means is crucial in statistical analysis.
• High-dimensional data ('large-p-small-n') presents challenges for traditional methods.
• Existing Bayesian methods struggle with large covariance matrices in high dimensions.

Purpose of the Study:

• To develop a robust Bayes factor testing procedure for high-dimensional data.
• To address limitations of existing methods in 'large-p-small-n' settings.
• To propose a novel Bayesian approach using random projections.

Main Methods:

• Utilizing lower-dimensional random projections of high-dimensional data vectors.
• Employing a restricted most powerful Bayesian test (RMPBT) by optimizing the prior.
• Constructing a test statistic from an ensemble of Bayes factors from multiple random projections.

Main Results:

• The proposed test is shown to be unbiased.
• The test demonstrates local consistency under mild conditions.
• The method's efficacy is validated through simulations and real-world data.

Conclusions:

• The random projection-based Bayes factor offers a viable solution for high-dimensional mean comparison.
• This approach circumvents the need for large covariance matrix estimation.
• The RMPBT provides a powerful and consistent Bayesian test in challenging data settings.