  Score-based methods for learning Bayesain networks(BN) aim to maximizing the
global score functions. However, if local variables have direct and indirect
dependence simultaneously, the global optimization on score functions misses
edges between variables with indirect dependent relationship, of which scores
are smaller than those with direct dependent relationship. In this paper, we
present an identifiability condition based on a determined subset of parents to
identify the underlying DAG. By the identifiability condition, we develop a
two-phase algorithm namely optimal-tuning (OT) algorithm to locally amend the
global optimization. In the optimal phase, an optimization problem based on
first-order Hilbert-Schmidt independence criterion (HSIC) gives an estimated
skeleton as the initial determined parents subset. In the tuning phase, the
skeleton is locally tuned by deletion, addition and DAG-formalization
strategies using the theoretically proved incremental properties of high-order
HSIC. Numerical experiments for different synthetic datasets and real-world
datasets show that the OT algorithm outperforms existing methods. Especially in
Sigmoid Mix model with the size of the graph being ${\rm\bf d=40}$, the
structure intervention distance (SID) of the OT algorithm is 329.7 smaller than
the one obtained by CAM, which indicates that the graph estimated by the OT
algorithm misses fewer edges compared with CAM.Source code of the OT algorithm
is available at https://github.com/YafeiannWang/optimal-tune-algorithm.
