Estimating from No Data: Deriving a Continuous Score from Categories

A method is presented for deriving a continuous numerical score from a model trained only on categorical data. The technique addresses creating fine-grained predictions, like a patient's risk level, when available labels are discrete outcomes such as "recovered," "hospitalized," or "died." The proposed solution involves a neural network with a bottleneck architecture, using a low-capacity linear model to produce a single score. A special "category approximator head" is then added, which learns the thresholds between the categories and allows the model to be trained using standard categorical cross-entropy loss. This approach enables the network to output a continuous, interpretable score while learning from discrete labels.