## 1 Introduction  
The advent of the Internet and the subsequent era of big data have led to an overwhelming influx of information. Coping with the massive expansion of data has become increasingly challenging, making it arduous for users to extract useful information from this abundance [1]. To address the issue of information overload, the recommendation system has emerged as a solution. The primary objective of the recommendation system is to provide improved predictive outcomes by leveraging a user's historical behavior data, analyzing user preferences, and generating product recommendations based on similarity. Collaborative filtering technology serves as the foundation for most contemporary recommendation systems [2]. The fundamental concept underlying this approach is that users with similar preferences tend to make analogous choices in the future. The implementation of collaborative filtering algorithms commonly revolves around matrix decomposition. This technique involves decomposing the user-item rating matrix into smaller-dimensional matrices: a user preference matrix and an item attribute matrix. The user's predicted rating for an item is then derived by computing the dot product of these two matrices. Within the recommendation system, numerous biases emerge, including but not limited to selection bias, position bias, and popularity bias [3]. These biases undermine the system's ability to accurately discern user preferences, consequently compromising the efficacy of the recommendation process [4; 5; 6]. The accuracy of recommendations serves as a crucial metric for evaluating the performance of recommendation systems. A higher level of accuracy directly correlates with improved system efficiency. In the task of predicting recommendation scores, the primary performance indicator is the recommendation accuracy, which measures the extent to which the predicted scores align with the actual scores, specifically by minimizing the discrepancy between them. This paper aims to address the issue of selection bias in recommendation systems. Selection bias refers to the phenomenon where users have the freedom to select which products they rate, resulting in a propensity to rate their favored products more frequently. Consequently, the collected ratings fail to represent the true ratings of the products, leading to the missing not at random (MNAR) problem and resulting in missing rating data [7; 8; 9]. However, existing studies often assume that the rating data is missing at random (MAR), which is not realistic in practice. Since user-item interaction data in recommender systems is observational rather than experimental, selection bias is present in nearly all data. For instance, in a movie recommendation system, users tend to watch and rate movies they like while seldom providing ratings for movies they are uninterested in [10]. The inverse propensity score (IPS) is a method that offers an effective means to mitigate bias in recommender systems [11]. However, the algorithm's ability to eliminate bias primarily relies on the accuracy of propensity score estimation. In the context of selection bias, past approaches primarily estimate propensity scores based on the marginal probability of users rating items, which inherently possess certain limitations. The primary contributions of this paper are as follows:  
1. We address the issue of selection bias in recommendation systems by developing a propensity score estimation method called the Selection bias Propensity score Estimator. This method takes into account both user and item information, providing a comprehensive approach to eliminating selection bias. 2. To construct a recommendation model that effectively removes selection bias, we integrate the SPE method with two traditional matrix factorization-based recommendation algorithms. We additionally introduce inverse weighting of the propensity score into the loss function, enabling the model to mitigate the impact of selection bias during the recommendation process. 3.