When we try to make inferences or generalizations in response to our observations of the external world based on our own internal knowledge built out of previous experiences, there is always an implicit band of confidence by which we make our inferences. The degree of certainty depends on the similarity of the contextual details surrounding our observation to those that have developed our internal knowledge previously. If we observe a moon-like shape in the night sky, we would make an inference for what it is spontaneously with almost zero uncertainty, but if our eyes catch a moving geometrical shape that resembles none of any known night-sky objects we are accustomed to, we would be tempted to make an inference, e.g. that it’s an UFO yet with much higher degree of uncertainty. Are the predictions of artificial neural networks also accompanied by uncertainty estimates? By virtue of their mathematical definitions, there is no natural candidate structure within a neural network’s architecture that will supply such an information. 

Deep Evidential learning is a recent proposal that fills up this gap. In contrast to a conventional neural network, a Deep Evidential model yields as model outputs, (mean) prediction values coupled with their uncertainty distributions, and arguably, this is a much healthier artificial intelligent system that can at least express its own uncertainty when making predictions for new data based on weights trained upon previous data. More technically, this framework equips a generic neural network with an ability to express uncertainty estimates of its predictions by asserting the existence of a Bayesian prior distribution for model predictions right from the outset. Upon completion of network training, a Deep Evidential model will yield predictions together with a distribution of uncertainty values that reflect how ‘certain’ the model is in making predictions. We should mention that Deep Evidential learning belongs to the broad class of uncertainty quantification frameworks that have received increasing attention in the machine learning community. There are two major variants of this model:

• Deep Evidential Regression by Amini et al. 
• Deep Evidential Classification by Sensoy et al. 

which are devoted to the tasks of regression and classification respectively. In this work, we reformulated the loss function that was proposed in the work by Amini et al. so that it can be implemented smoothly for radiotherapy dose prediction with the input objects being CT images with various organs-at-risk and target masks ( brainstem, esophagus, parotids, larynx, mandible, spinal cord and radiation target regions receiving 56 Gy, 63 Gy and 70 Gy ). Our medical image dataset is drawn from the Open Knowledge-Based Planning Challenge AAPM 2020 which is the hitherto largest international effort in gathering a unified single open dataset for deep learning models in the context of radiotherapy dose prediction. 

We found that there was a high degree of correlation between model errors and the uncertainty estimates, so this implies that we can use the elevated regions of model uncertainty to discover and highlight potential regions of model errors. As an application, we can use the knowledge of model uncertainty to draw statistical confidence bands for the Dose-Volume-Histogram which is a common tool for radiation oncologists to analyze various aspects of the planned radiotherapy treatment.

In the Figure below, we plot the Dose-Volume-Histogram for an example patient in the AAPM dataset who was given 6MV photon radiotherapy. The horizontal axis represents the radiation dose imparted to some specific part of the patient’s body (see the legend for each anatomical part; note that PTV70 represents the planning target volume intended to receive 70 Gy of radiation dose, etc. ), whereas the vertical axis represents the percentage volume of the anatomical site receiving a dose equal or greater than the dose coordinate. The solid curves represent the dose predicted by our deep learning model whereas the dotted curves depict the actual dose distributions. The light shaded bands represent the 95% confidence bands that were part of our deep learning model’s output. Roughly speaking, the wider a band is, the less reliable the model’s prediction.