  Assuming a banded structure is one of the common practice in the estimation
of high-dimensional precision matrix. In this case, estimating the bandwidth of
the precision matrix is a crucial initial step for subsequent analysis.
Although there exist some consistent frequentist tests for the bandwidth
parameter, bandwidth selection consistency for precision matrices has not been
established in a Bayesian framework. In this paper, we propose a prior
distribution tailored to the bandwidth estimation of high-dimensional precision
matrices. The banded structure is imposed via the Cholesky factor from the
modified Cholesky decomposition. We establish the strong model selection
consistency for the bandwidth as well as the consistency of the Bayes factor.
The convergence rates for Bayes factors under both the null and alternative
hypotheses are derived which yield similar order of rates. As a by-product, we
also proposed an estimation procedure for the Cholesky factors yielding an
almost optimal order of convergence rates. Two-sample bandwidth test is also
considered, and it turns out that our method is able to consistently detect the
equality of bandwidths between two precision matrices. The simulation study
confirms that our method in general outperforms the existing frequentist and
Bayesian methods.
