Quantifying Epistemic and Aleatoric Uncertainty in 3D U-Net Segmentation

This work shows a derivation of a multinomial probability function and quantitative measures of the data and epistemic uncertainty as direct output of a 3D U-Net segmentation network. A set of T1 brain MRI images were downloaded from the Connectome Project and segmented using FMRIB9s FAST algorithm to be used as ground truth. A 3D U-Net neural network was trained with sample sizes of 200, 500, and 898 T1 brain images using a loss function defined as the negative logarithm of the likelihood based on a derivation of the definition of the multinomial probability function. From this definition, the epistemic (model) and aleatoric (data) uncertainty equations were derived and used to quantify maps of the uncertainty in data prediction. The epistemic and aleatoric uncertainty decreased based on the increasing number of training data used to train the neural network. The neural network trained with 898 volumes resulted in uncertainty maps that were high primarily in the tissue boundary regions. The uncertainty was averaged over all test data (connectome and tumor separately) and the epistemic uncertainty showed a decreasing trend, as expected, with increasing numbers of data used to train the model. The aleatoric uncertainty showed a similar trend, but it was less obvious, which was also expected as the aleatoric uncertainty is not expected to be as dependent on the number of training data. The derived data and epistemic uncertainty equations from a multinomial probability distribution are applicable for all 2D and 3D neural networks.