Fast sampling with Gaussian scale-mixture priors in high-dimensional regression.

Area of Science:

• Statistics
• Computational Statistics
• Machine Learning

Background:

• Sampling from multivariate Gaussian distributions is crucial in statistical modeling.
• Existing methods, like Cholesky factorization, have cubic computational complexity, limiting scalability for high-dimensional problems.
• Gaussian scale mixture priors are increasingly used in high-dimensional settings.

Purpose of the Study:

• To propose a novel, computationally efficient algorithm for sampling from structured multivariate Gaussian distributions.
• To reduce the computational complexity of sampling compared to existing methods.
• To demonstrate the algorithm's applicability in high-dimensional statistical modeling.

Main Methods:

• The proposed algorithm utilizes matrix multiplications and linear system solutions.
• It avoids computationally intensive Cholesky factorizations.
• Computational complexity scales linearly with the data dimension.

Main Results:

• Achieved linear computational complexity with respect to the dimension of the distribution.
• Demonstrated effectiveness in a high-dimensional regression problem using a horseshoe prior.
• The algorithm is broadly applicable to models employing Gaussian scale mixture priors.

Conclusions:

• The new algorithm provides a significant computational advantage for sampling from structured multivariate Gaussians.
• It enables efficient analysis in high-dimensional statistical applications.
• The method is a valuable tool for researchers working with complex prior structures.