Estimating large precision matrices via modified Cholesky decomposition

Abstract

We introduce the k-banded Cholesky prior for estimating a high-dimensional bandable precision matrix via the modified Cholesky decomposition. The bandable assumption is imposed on the Cholesky factor of the decomposition. We obtained the P-loss convergence rate under the spectral norm and the matrix ∞ norm and the minimax lower bounds. Since the P-loss convergence rate (Lee and Lee (2017)) is stronger than the posterior convergence rate, the rates obtained are also posterior convergence rates. Furthermore, when the true precision matrix is a k 0 -banded matrix with some finite k 0 , the obtained P-loss convergence rates coincide with the minimax rates. The established convergence rates are slightly slower than the minimax lower bounds, but these are the fastest rates for bandable precision matrices among the existing Bayesian approaches. A simulation study is conducted to compare the performance to the other competitive estimators in various scenarios.

Minimax posterior convergence rates and model selection consistency in high-dimensional DAG models based on sparse Cholesky factors

“…Proof. We follow closely the line of the proof of Lemma 7.4 in Lee and Lee (2017). Let Ω 0n ∈ U * p and N S 0 ,ǫ 0 be the set defined at Lemma 7.2.…”

“…Note that a decomposable graph can be converted to a perfect DAG, a special case of the DAGs, by ignoring directions. Lee and Lee (2017) obtained the posterior convergence rates and minimax lower bounds for the precision matrices, but only bandable Cholesky factors were considered. Posterior convergence rates for the precision matrices as well as strong model selection consistency were recently derived by Cao, Khare and Ghosh (2017) for sparse DAG models.…”

Bayesian Bandwidth Test and Selection for High-dimensional Banded Precision Matrices

“…Since we have bandwidth selection consistency (Theorem 3.1), we suggest using the posterior mode to estimate the true bandwidth k 0 . We chose the bandwidth test of An et al (2014) as a frequentist competitor and the bandwidth selection procedures of Banerjee and Ghosal (2014) and Lee and Lee (2017) as Bayesian competitors. Significance levels for bandwidth tests in An et al (2014) were varied α = 0.001, 0.005, 0.01, but only the result with α = 0.01 are reported since they gave similar results.…”

“…They derived the posterior convergence rate of the precision matrix under the G-Wishart prior (Roverato;2000). Lee and Lee (2017) considered a similar class to that of Banerjee and Ghosal (2014), but assumed bandable Cholesky factors instead of bandable precision matrices. They showed the posterior convergence rates of the precision matrix as well as the minimax lower bounds.…”

“…The key difference from Lee and Lee (2017) is on the choice of prior distributions which will be introduced in Section 2.3. In addition, we focus on bandwidth selection and tests, while Lee and Lee (2017) mainly studied the convergence rates of the precision matrix for a given or fixed bandwidth.…”