I believe that in most cases, the positive case is arbitrarily defined to be the case with a lower frequency. If I instead define the negative case as the one with the lower frequency, what differences would this entail for the model?
For all of the algorithms/procedures that I know, it makes no difference between a positive or negative interpretation of a class.
The fact that one category has a lower frequency (by that I am assuming that you mean a low probability) is insignificant. It is just a matter of communication, by maintaining the general designations in a specific field so that others will be able to read your work more easily.
As an example, in anomaly detection, one designates the cases with lower frequency as anomaly, but even so, technically it does not matter how you label your cases, they are interchangeable.