In a reliability or maintenance analysis of a complex system, it is important to be able to identify the main causes of failure. Therefore a Weibull competing risk model is generally used. However, in this framework estimating the model parameters is a difficult ill-posed problem. Indeed, the cause of the system failure may not be identified and may also be censored by the duration of the study. In addition, the other causes are naturally censored by the first one. In this paper, we propose a new method for estimating the parameters of the Weibull competing risk model, with masked causes and heavily censored data. We use a Bayesian restoration of missing data through a Bayesian importance sampling of parameters with a weakly informative prior distribution. The mode of the posterior distribution can thus be obtained directly or in a approximate way. The proposed method is not an iterative method and therefore is not costly in terms of computing time. Experiments based on simulated data and a reliability data set show that the prediction performance of the proposed method is superior to the maximum likelihood method, the standard EM algorithm and the Gibbs sampler, for low to very heavy censoring rates and even in the case of weakly separated components.