I’m working on a multilabel classification problem.  In my current version, I have four potential labels: “Hospitalized”, “Intubated”, “Deceased”, and “Pneumonia”.  My model trains well and has provided some interesting insights on cases with at least one label, but none of my post-training analyses account for the cases that have NO labels, i.e. patients that contracted the disease but recovered at home with no complications.
I started trying to modify my post training analysis routines to add in a 5th “None” label, but quickly realized this wasn’t going to work.  What was my predicted likelihood of “None”?  Is it (1 - [sum of other label likelihoods])?  Not really…  This value is often negative.  At that point, I considered adding in a 5th “None” label before training and revising my multihot target matrices accordingly, but I wasn’t sure that was the right way to go either given that the truth value of that 5th label can be easily calculated given the truth values for the other four labels.
So my questions are two fold:

Is adding in the 5th “None” label prior to training the right way forward?
How do interdependencies between labels (i.e. label E is always 0 if label A, B, C, or D is 1 and always 1 otherwise) impact multilabel classification problems?