Expert knowledge holds a pivotal role in artificial intelligence models. Constrained by the subjectivity and ignorance of human cognition, it is imperfectly reliable. Modeling and decision-making driven by such knowledge may generate large risks. To this end, it is necessary to investigate a mechanism for handling such imperfectly reliable knowledge. In this paper, the reliability of knowledge is described as expert reliability. A novel rule-based modeling framework with expert reliability is proposed correspondingly, including the following four parts: modeling, reasoning, optimization and robustness analysis. The main works are: (1) Based on the transparent knowledge representation of belief rule base (BRB), a linguistic Z-number BRB (LZ-BRB) is proposed, where the linguistic Z-number quantitatively represents expert reliability. (2) An improved evidential reasoning (ER) rule is developed to obtain the inference result of the LZ-BRB model. (3) A data-driven parameter optimization model is designed to reduce modeling errors caused by imperfectly reliable knowledge. (4) The robustness analysis of expert reliability is performed to further analyze its influence on the inference result. Finally, a fiber optic gyro (FOG) health evaluation case verifies the proposed method.