## Bayesian Analysis

### Fast Model-Fitting of Bayesian Variable Selection Regression Using the Iterative Complex Factorization Algorithm

#### Abstract

Bayesian variable selection regression (BVSR) is able to jointly analyze genome-wide genetic datasets, but the slow computation via Markov chain Monte Carlo (MCMC) hampered its wide-spread usage. Here we present a novel iterative method to solve a special class of linear systems, which can increase the speed of the BVSR model-fitting tenfold. The iterative method hinges on the complex factorization of the sum of two matrices and the solution path resides in the complex domain (instead of the real domain). Compared to the Gauss-Seidel method, the complex factorization converges almost instantaneously and its error is several magnitude smaller than that of the Gauss-Seidel method. More importantly, the error is always within the pre-specified precision while the Gauss-Seidel method is not. For large problems with thousands of covariates, the complex factorization is 10–100 times faster than either the Gauss-Seidel method or the direct method via the Cholesky decomposition. In BVSR, one needs to repetitively solve large penalized regression systems whose design matrices only change slightly between adjacent MCMC steps. This slight change in design matrix enables the adaptation of the iterative complex factorization method. The computational innovation will facilitate the wide-spread use of BVSR in reanalyzing genome-wide association datasets.

#### Article information

Source
Bayesian Anal., Volume 14, Number 2 (2019), 573-594.

Dates
First available in Project Euclid: 29 August 2018

https://projecteuclid.org/euclid.ba/1535508388

Digital Object Identifier
doi:10.1214/18-BA1120

Mathematical Reviews number (MathSciNet)
MR3934098

Zentralblatt MATH identifier
07045443

#### Citation

Zhou, Quan; Guan, Yongtao. Fast Model-Fitting of Bayesian Variable Selection Regression Using the Iterative Complex Factorization Algorithm. Bayesian Anal. 14 (2019), no. 2, 573--594. doi:10.1214/18-BA1120. https://projecteuclid.org/euclid.ba/1535508388

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#### Supplemental materials

• Fast model-fitting of Bayesian variable selection regression using the iterative complex factorization algorithm (Supplementary). The online appendix includes additional numerical studies (S1–S3), a summary of the MCMC algorithm (S4), examples for updating the Cholesky decomposition (S5) and a proof for the stationary distribution of the exchange algorithm (S6). S1 compares the performance of the iterative methods using simulated datasets; S2 studies the behavior of the convergence rates of the iterative methods; S3 provides numerical evidence for the assumption $\eta_min^{2} \approx 0$ used in Section 2.3.