A new hybrid copula-based nonparametric Bayesian model for risk assessments of water inrush

Bayesian networks (BNs) can be automatically constructed with field data when the data can sufficiently support the objectivity of the model. However, in most risk assessments, field data cannot effectively support learning with BNs. In this paper, a new hybrid method is proposed to construct BNs and estimate the corresponding parameters considering the objectivity of field data and the accessibility of expert knowledge. This method is combined with the ISM-K2 (interpretive structural model) algorithm, copula theory, and the nonparametric method. First, the ISM is used to identify the relationships among the directly and indirectly related variables (i.e., obtain the parent variable set). Second, based on the parent variable set, the K2 algorithm is used to construct BNs with the search volume reduced from an exponential to a quadratic form. Third, copula theory is introduced to consider several marginal distributions of variables, and a copula parameter is used to replace the multivariate joint cumulative distribution. The Gumbel copula function is first introduced to replace the often-used normal copula function. Fourth, four types of distributions are utilized to fit the characteristics of the variables as the marginal distribution by using a nonparametric method. Finally, the proposed method was used to construct BNs for water inrush and estimate the risk of water inrush for a tunnel.