Shannon entropy and Fisher information are pivotal in the information theory area. The presence of outliers in data and using an inappropriate model may cause misleading inferential results in the amount of information. Our aim of this paper is to compute the amount of Shannon entropy and Fisher information that exists in the Pareto distribution in the presence of multiple outliers. Unlike the existing methods in the literature, we present a good method for the estimation of Shannon entropy and Fisher information to cope with the allowing for the possibility of outliers. In this regard, we focus on the Bayesian approach proposed by [32] based on the contaminated Pareto distribution. We implement the Gibbs sampler which is a simple and rational method for computing Bayesian estimation of Shannon entropy and Fisher information under different loss functions. Some simulation studies are conducted to investigate the performance of the proposed methodology under various sample sizes and the number of outliers. In the end, two examples of real insurance claim data are studied to illustrate the superiority of the proposed model in analyzing datasets and computing the amount of Shannon entropy and Fisher information.

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