# An All-Batch Loss for Constructing Prediction Intervals
## 4 Discussions  
Case study 1 demonstrates that the model trained by the AB loss obtains higher quality PIs under samples of all batch sizes if compared with the QD loss. According to Table 3, the model obtained by QD loss does not fit the superimposed noise well for batch sizes of 16 and 32. The performance of the QD loss improves gradually with the increasing of batch size for training. This implies that the QD loss may be inadaptable to small sample sets, while the AB loss performs well for all batch sizes. The AB loss takes advantage of the direct application of the Taylor expansion to approximation, without the necessity of changing the basic assumption of the binomial distribution, which allows it to solve more practical problems while ensuring accuracy.  
Many traditional PI construction methods were built based on the various assumptions regarding data distribution, which may be unsatisfied in real-world datasets. However, as a distribution-free method, the AB loss is more suitable for practical applications. As shown in Figure 4, compared with traditional methods such as MVE, more reasonable results may be obtained when different noises are added to the dataset. Thus, there is no limitation on data distribution when utilizing the AB loss for model training. The results of numerical experiments also prove that the AB loss can obtain higher quality PIs than existing state-of-the-art methods in most cases.  
In addition, it can be seen that the AB loss is more sensitive to confidence measurement than the QD loss, which indicates that the PIs generated by the proposed method are more likely to meet the required confidence level. When utilizing the AB loss, more energy should be devoted to sharpness measurement, which may result in high quality PIs. Meanwhile, from the curves of CWC with weight parameters in Figures 5 and 6, it can be seen that the best parameters are easier to find when adopting AB loss, which simplifies the parameter adjustment process and makes it more convenient to deploy in practice.  
In summary, the results prove the reliability and the quality of PIs constructed by the AB loss. By introducing the AB loss into the PI generation model, it is possible to use small batches for model training, which is appropriate in many real-world tasks.  
Future works may be investigated in the following aspects:  
1. The realization of the AB loss in this study did not consider the impact of model uncertainty. Since two networks are used to predict the prediction estimate and the bias estimate, the uncertainty of the model was not determined in the implementation. It is possible to use methods such as bootstrap to estimate model uncertainty to improve the quality of the results;
2. The existence of the penalty term means it is not necessary to initialize the model before the training process, although this feature increases the computational complexity. More efforts should be made to simplify the penalty term;
3. It is difficult to adjust the weight parameter of the AB loss on different datasets. A more efficient way should be explored to balance the contradiction between confidence and sharpness.