The BI_RADS score is a probabilistic reporting tool used by radiologists to express the level of uncertainty in predicting breast cancer based on some morphological features in mammography images. There is a significant variability in describing masses which sometimes leads to BI_RADS misclassification. Using a BI_RADS prediction system is required to support the final radiologist decisions. In this study, the uncertainty information extracted by a Bayesian deep learning model is utilized to predict the BI_RADS score. The investigation results based on the pathology information demonstrate that the f1-scores of the predictions of the radiologist are 42.86%, 48.33% and 48.28%, meanwhile, the f1-scores of the model performance are 73.33%, 59.60% and 59.26% in the BI_RADS 2, 3 and 5 dataset samples, respectively. Also, the model can distinguish malignant from benign samples in the BI_RADS 0 category of the used dataset with an accuracy of 75.86% and correctly identify all malignant samples as BI_RADS 5. The Grad-CAM visualization shows the model pays attention to the morphological features of the lesions. Therefore, this study shows the uncertainty-aware Bayesian Deep Learning model can report his uncertainty about the malignancy of a lesion based on morphological features, like a radiologist.