  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
$\ell_{\infty}$ 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.
