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Recently, deep learning has achieved remarkable successes in medical image analysis. Although deep neural networks generate clinically important predictions, they have inherent uncertainty. Such uncertainty is a major barrier to report these predictions with confidence. In this paper, we propose a novel yet simple Bayesian inference approach called SoftDropConnect (SDC) to quantify the network uncertainty in medical imaging tasks with gliomas segmentation and metastases classification as initial examples. Our key idea is that during training and testing SDC modulates network parameters continuously so as to allow affected information processing channels still in operation, instead of disabling them as Dropout or DropConnet does. When compared with three popular Bayesian inference methods including Bayes By Backprop, Dropout, and DropConnect, our SDC method (SDC-W after optimization) outperforms the three competing methods with a substantial margin. Quantitatively, our proposed method generates substantial improvements in prediction accuracy (by 3.4%, 2.5%, and 6.7% respectively for whole tumor segmentation in terms of dice score; and by 11.7%, 3.9%, and 8.7% respectively for brain metastases classification) and greatly reduced epistemic and aleatoric uncertainties. Our approach promises to deliver better diagnostic performance and make medical AI imaging more explainable and trustworthy.