My notes define the false positive rate as
$$\dfrac{\text{false positives}}{\text{true negatives} + \text{false positives}} = \dfrac{\text{false positives}}{\text{total negatives}}$$
and the false negative rate as
$$\dfrac{\text{false negatives}}{\text{false negatives} + \text{true positives}} = \dfrac{\text{false negatives}}{\text{total positives}}$$
It then says the following:
We typically cannot control both these errors:
- If we change parameters in the classifier that make one smaller, then the other one often gets larger.
I don't really see why this must mathematically be true. It seems to me that these two rates depend on the efficacy of the technology, and that there is no mathematical reason why both cannot be improved at the same time. Am I misunderstanding something, or is what is stated in the notes incorrect?