Automatic classification of diabetic retinopathy from retinal images has been increasingly studied using deep neural networks with impressive results. However, there is clinical need for estimating uncertainty in the classifications, a shortcoming of modern neural networks. Recently, approximate Bayesian neural networks (BNNs) have been proposed for this task, but previous studies have only considered the binary referable/non-referable diabetic retinopathy classification applied to benchmark datasets. We present novel results for 9 BNNs by systematically investigating a clinical dataset and 5-class classification scheme, together with benchmark datasets and binary classification scheme. Moreover, we derive a connection between entropy-based uncertainty measure and classifier risk, from which we develop a novel uncertainty measure. We observe that the previously proposed entropy-based uncertainty measure improves performance on the clinical dataset for the binary classification scheme, but not to such an extent as on the benchmark datasets. It improves performance in the clinical 5-class classification scheme for the benchmark datasets, but not for the clinical dataset. Our novel uncertainty measure generalizes to the clinical dataset and to one benchmark dataset. Our findings suggest that BNNs can be utilized for uncertainty estimation in classifying diabetic retinopathy on clinical data, though proper uncertainty measures are needed to optimize the desired performance measure. In addition, methods developed for benchmark datasets might not generalize to clinical datasets.