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One of the major challenges in modern day statistics is to formulate models and develop inferential procedures to understand the complex multivariate relationships present in high-dimensional datasets. In this paper, we address the issue of model determination for DAGs, with respect to a given ordering of the variables, together with the corresponding parameter estimation. For this, we use a hierarchical mixture prior and develop a Gibbs sampling algorithm to carry out the posterior computations. We first focus on the Gaussian DAG models and calculate the posterior probability of being the edge between two nodes. We then extend our idea to construct a DAG for discrete data under the assumption that the data generated by discretization of the marginal distributions of a latent multivariate Gaussian distribution via a set of predetermined threshold values. Results show that the proposed method has high accuracy. The source code is available at [ABSTRACT FROM AUTHOR]