Non-Gaussian processes are frequently encountered in engineering problems, posing a challenge when it comes to identification. The main challenge in the identification arises from the fact that a non-Gaussian process can be treated as a collection of infinite dimensional non-Gaussian variables. The application of the hierarchical Bayesian modeling (HBM) framework is constrained due to the inherent complexity of dimensionality and non-Gaussian characteristics associated with these variables. To tackle the issue of dimensionality, the improved orthogonal series expansion (iOSE) representing a non-Gaussian process by time functions with non-Gaussian coefficients, which are readily obtained from discretizing the process at some specific time points, is introduced within the HBM framework. In particular, the iOSE is embedded to convert the identification of a non-Gaussian process into a finite number of non-Gaussian coefficients. Regarding their non-Gaussian characteristics, polynomial chaos expansion (PCE) is used to quantify the uncertainty of the non-Gaussian coefficients with parameters in PCE treated as hyper parameters to be estimated by the HBM framework. The proposed framework is applicable to the identification of both stationary and nonstationary non-Gaussian processes. The effectiveness of non-Gaussian process quantification by the proposed framework is demonstrated using simulated data of a non-stationary extreme value process. The applicability of this approach for non-Gaussian process identification is validated by accurately identifying a stochastic load in a structural dynamic problem. Furthermore, it is successfully applied to the reconstruction of random mode shapes of a building arising from different environmental conditions.