## 1 Introduction  
In the traditional design methodologies of mechanical systems, design variables are optimized to maximize the performances based on deterministic assumption [1, 2]. However, there are a lot of uncertainties in real mechanical systems, including input loads, product dimensions, material properties, and manufacturing tolerance [3, 4, 5, 6]. To quantitatively represent these uncertainties in actual conditions and analyze their influence on performance functions, plenty of uncertainty analysis methods have been proposed in the design of mechanical systems. Initially, the uncertain input variables are assumed to be aleatoric uncertainties. By means of low-cost experiment or rapid simulation analysis, the uncertainty information can be acquired, meanwhile the distribution types and distribution parameters are treated as determinate and accurate variables. Many probabilistic uncertainty analysis methodologies have been proposed to deal with aleatoric uncertainties, such as uncertainty quantification based on available uncertainty information [7, 8, 9, 10], uncertainty propagation among multiple variables [11, 12], probabilistic sensitivity analysis [13, 14], probabilistic uncertainty optimization design [15, 16, 17], etc. However, under some circumstances, it is time-consuming and complicated to acquire enough uncertainty information to construct deterministic representation functions for uncertain input variables. Therefore, many representation methods of epistemic uncertainties have been employed, which include possibility theory [18], probability box [19, 20], evidence theory [21, 22, 23], fuzzy set [24, 25] and convex models [26, 27]. [PERSON] and [PERSON] [28, 29] proposed an uncertainty representation method based on sparse points and/or interval data, the distribution type uncertainty was quantified by using Bayesian model averaging and Bayesian hypothesis testing approaches, while the uncertainties of distribution parameters were calculated using the maximum likelihood estimation methodology. [PERSON] et al. [30; 31] estimated the distribution parameters of uncertainty variables using a worst-case maximum-likelihood-based approach, and developed a robustness-based design optimization methodology under aleatoric and epistemic uncertainties. [PERSON] [32] identified the parameter uncertainties using a stochastic inverse algorithm based on Karhunen-Loeve expansion and the maximum likelihood estimation method. [PERSON] [33; 34] derived a complete probabilistic expression of epistemic uncertainty according to statistical information of uncertain variables, and applied the proposed cubic-normal distribution to a variety of areas in reliability engineering. [PERSON] [35; 36] developed the state-dependent parameter method to calculate the sensitivity indices of uncertainties of input variables and their distribution parameters. [PERSON] [37; 38] estimated the uncertainty of the performance function using the maximum entropy and univariate dimension reduction methods based on multimodal probability density functions of input uncertainty variables. [PERSON] [39] developed a Monte Carlo framework to deal with aleatoric and epistemic uncertainty, and computational approaches were outlined for propagation and sensitivity analysis with distribution parameter uncertainty. [PERSON] [40] developed a non-intrusive imprecise probabilistic integration framework to quantify uncertainties of epistemic variables with uncertain distribution parameters. The uncertainty information of epistemic uncertainties in the above methods comes from single sources. However, in order to decrease the error from single source and increase the available uncertainty information, the uncertainty data for epistemic uncertainties may come from multiple different sources in some conditions, such as cases involving multiple sensors or various expert knowledge.