In recent years, the involvement of synthetic strongly labeled data,weakly labeled data and unlabeled data has drawn much research attentionin semi-supervised sound event detection (SSED). Self-training models carry out predictions without strong annotations and then take predictions with high probabilities as pseudo-labels for retraining. Such models have shown its effectiveness in SSED. However, probabilities are poorly calibrated confidence estimates, and samples with low probabilities are ignored. Hence, we introduce a method of learning confidence deliberately and retaining all data distinctly by applying confidence as weights. Additionally, linear pooling has been considered as a state-of-the-art aggregation function for SSED with weak labeling. In this paper, we propose a power pooling function whose coefficient can be trained automatically to achieve nonlinearity. A confidencebased semi-supervised sound event detection (C-SSED) framework is designed to combine confidence and power pooling. The experimental results demonstrate that confidence is proportional to the accuracy of the predictions. The power pooling function outperforms linear pooling at both error rate and F1 results. In addition, the C-SSED framework achieves a relative error rate reduction of 34% in contrast to the baseline model.