In automatic emotion recognition (AER), labels assigned by different human annotators to the same utterance are often inconsistent due to the inherent complexity of emotion and the subjectivity of perception. Though deterministic labels generated by averaging or voting are often used as the ground truth, it ignores the intrinsic uncertainty revealed by the inconsistent labels. This paper proposes a Bayesian approach, deep evidential emotion regression (DEER), to estimate the uncertainty in emotion attributes. Treating the emotion attribute labels of an utterance as samples drawn from an unknown Gaussian distribution, DEER places an utterance-specific normal-inverse gamma prior over the Gaussian likelihood and predicts its hyper-parameters using a deep neural network model. It enables a joint estimation of emotion attributes along with the aleatoric and epistemic uncertainties. AER experiments on the widely used MSP-Podcast and IEMOCAP datasets showed DEER produced state-of-the-art results for both the mean values and the distribution of emotion attributes.