## 4 Experiment
### Experimental Results and Discussion  
The score prediction results of the proposed LFM-SPE model and BiasSVD-SPE model, along with other comparative models, are presented in Tables 2 and 3. By analyzing the experimental results, the following conclusions can be drawn.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Data Set** & **BiasSVD** & **LFM** & **LFM-IPS** & **LFM-DR** & **LFM-MRDR** & **BiasSVD-SPE** & **LFM-SPE** \\ \hline ML 100K & 0.826 & 0.810 & 0.805 & 0.801 & 0.797 & 0.803 & 0.795 \\ ML 1M & 0.771 & 0.749 & 0.745 & 0.738 & 0.733 & 0.746 & 0.736 \\ ML latest & 0.991 & 0.988 & 0.980 & 0.974 & 0.971 & 0.973 & 0.965 \\ \hline Average & 0.862 & 0.849 & 0.843 & 0.838 & 0.834 & 0.841 & 0.832 \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of RMSE experimental results of each model.  
Figure 3: The impact of recommendation threshold parameter \(\mathbf{\alpha}\) on the recommendation performance of LFM-SPE model. (**a**) Impact of \(\mathbf{\alpha}\) on the evaluation index RMSE and (**b**) impact of \(\mathbf{\alpha}\) on the evaluation index MAE.  
The BiasSVD-SPE model outperforms the BiasSVD matrix factorization algorithm, with an improvement of 2.44% and 3.03% in RMSE and MAE indicators, respectively. Unlike the traditional BiasSVD algorithm, which simply incorporates bias items to account for user and item deviations in score prediction, the BiasSVD-SPE model addresses selection bias by calculating propensity scores for each sample group. It effectively handles selection bias through inverse propensity score weighting, considering variations across different data groups. Consequently, the BiasSVD-SPE model exhibits a more substantial performance enhancement compared to the traditional BiasSVD algorithm as it comprehensively addresses data deviations and offers increased interpretability.  
The LFM-SPE model proposed in this paper demonstrates superior recommendation performance compared to other models. When compared to the traditional LFM model, the LFM-SPE model exhibits improvements of 2.00% and 2.97% in the RMSE and MAE evaluation indicators, respectively. Additionally, the recommendation performance of the LFM-IPS, LFM-DR, and LFM-MRDR models, which incorporate propensity scores, also shows enhancement. These findings highlight the significant impact of incorporating propensity scores to mitigate selection bias, leading to substantial improvements in the model's recommendation performance.  
Among the four LFM models with added propensity scores, the LFM-SPE model exhibited the highest recommendation performance. Specifically, the LFM-SPE model outperformed the LFM-IPS model, resulting in improvements of 1.30% and 1.91% in the RMSE and MAE evaluation indicators, respectively. This suggests that the SPE method is superior to IPS in eliminating selection bias. While the IPS algorithm estimates the propensity score based on the marginal probability of user-item selection, the proposed method in this paper simultaneously analyzes and leverages the user and item information, incorporating both the user's selection rate and the item's selection preference to estimate the propensity score. As a result, the propensity score estimate is more accurate, leading to improved recommendation performance.  
Furthermore, the LFM-SPE model demonstrated slightly better average recommendation performance across the three datasets compared to LFM-MRDR. This indicates that the LFM-MRDR model benefits from the effective reduction of variance through the MRDR double robust estimation method, which in turn mitigates the influence of bias on the recommendation algorithm. It is worth noting that the recommendation effect of LFM-MRDR surpassed that of LFM-DR, further highlighting the efficacy of the MRDR approach.  
Overall, the experimental results demonstrate that the LFM-SPE model, with its combined analysis of user and item information and accurate propensity score estimation, outperforms other LFM models with propensity scores in terms of recommendation performance.