## 4 Effectiveness of Data Imputation
### Observations  
Several interesting observations emerge from the heatmaps. First, the Signal-RL task demonstrates significantly greater sensitivity to imputation noise. This can be explained by the fact that sequential decision making is inherently more noise sensitive, combined with this task's use of \"general imputation\". In the heatmaps, red regions indicate where imputation improves model performance, while blue regions show where imputation is detrimental. These visualizations clearly reveal the trade-off between data missingness and noise introduction, with two distinct regions being identifiable:  
* **Imputation advantageous corner**: This region is located in the lower-right corner of the heatmap, where the data-missing ratio is high, and the imputation noise level is low. Accurate imputation in this region restores critical information, leading to significant improvements in model performance. * **Imputation disadvantageous edge**: This region is near the edge where \(q=1\). When the imputation noise level approaches 1.0, the noise introduced during imputation overwhelms the model, leading to performance degradation. Interestingly, the greatest harm occurs when the data-missing ratio is around \(p=0.6\). Additionally, the black dashed contour line corresponding to \(A(p,q)=0\) is overlaid on the heatmap, indicating the decision boundary between regions where imputation is advantageous versus disadvantageous. The black solid line shows the fitted decision boundary. The green and lime-green dashed lines denote the 68% and 95% confidence intervals, corresponding to \(\pm 1\) and \(\pm 1.96\) standard errors, respectively. A few notable differences between the Signal-RL and NLP-SL tasks can be observed:  
* **Signal-RL Decision Boundary**: The contour curve for the Signal-RL task lies much lower and is shifted to the right compared to the NLP-SL task. Moreover, the contour curve for Signal-RL is more ragged and fits to an exponential curve that starts at (0, 0) and intersects the line \(p=1\). * **NLP-SL Decision Boundary**: The contour line for the NLP-SL task is smoother and fits well to a logistic function. When \(p\) in range \([0,0.4]\), the decision boundary is relatively  
Figure 7: Heatmap of imputation advantage. The \(x\)-axis denotes the data-missing ratio \(p\), and the \(y\)-axis represents the imputation noise level \(q\). The black dashed line indicates the decision boundary, separating regions where imputation is beneficial from those where it is detrimental. The black solid line is the fitted curve of decision boundary. The green and lime-green dashed lines denote the 68% and 95% confidence intervals. (**a**) NLP-SL experiment: NLP-SL is a noise-insensitive task. The decision boundary lies above the diagonal and is well approximated by a logistic function. (**b**) Signal-RL experiment: Signal-RL exemplifies a noise-sensitive task. The contour line appears below the diagonal, more irregular in shape, and fits an exponential curve starting from (0, 0) and intersecting the line \(p=1\). stable and remains around \(q=0.68\). For \(p\) in range \([0.4,1.0]\), the contour line transitions into a sigmoid curve, with its midpoint around \(p=0.7\). This gradual transition reflects the trade-off between recovering critical information through accurate imputation and introducing ambiguities (e.g., incorrect word predictions) through noisy imputation. The sigmoid shape of the NLP-SL contour line reflects a smoother transition between advantageous and disadvantage regions as imputation noise increases, whereas the Signal-RL task exhibits an exponential drop-off in performance due to the compounding effect of errors in sequential decision making. Tasks can be classified based on their sensitivity to noise:  
* **Noise-sensitive tasks**: Tasks with contour curves below the diagonal (e.g., Signal-RL) are highly sensitive to noise, showing sharp performance degradation as imputation noise increases.