  Score-based methods have demonstrated their effectiveness in discovering
causal relationships by scoring different causal structures based on their
goodness of fit to the data. Recently, Huang et al. proposed a generalized
score function that can handle general data distributions and causal
relationships by modeling the relations in reproducing kernel Hilbert space
(RKHS). The selection of an appropriate kernel within this score function is
crucial for accurately characterizing causal relationships and ensuring precise
causal discovery. However, the current method involves manual heuristic
selection of kernel parameters, making the process tedious and less likely to
ensure optimality. In this paper, we propose a kernel selection method within
the generalized score function that automatically selects the optimal kernel
that best fits the data. Specifically, we model the generative process of the
variables involved in each step of the causal graph search procedure as a
mixture of independent noise variables. Based on this model, we derive an
automatic kernel selection method by maximizing the marginal likelihood of the
variables involved in each search step. We conduct experiments on both
synthetic data and real-world benchmarks, and the results demonstrate that our
proposed method outperforms heuristic kernel selection methods.
