Let's say I have three classes that represent the cervix type needed to be classified by my model. The overall goal is to predict the correct cervix class so health provider can give the patient the most appropriate treatment for their cervical cancer. Misclassifying cervix type would cost diagnosis time for health providers and treatment fee for the patients. In this case, is precision more important than recall?
Just to remember:
Recall: It's the ratio between the true positive and the false negative. So, it measures the ratio between the well predicted class and all the samples of this class. It's good when your goal is to identify all the samples of a class (for example, predict all people who have cancer).
Precision: It's the ratio between the true positive and the false positive. So, it measures the ratio between the well predicted class and all the samples that was predicted as this class. It's good when your goal is to not have false positive (for example, avoid predicting a woman is pregnant when actually she isn't).
In this case, I think it is important to know more about the consequences of a mistake. Is there some life risk if a person is of type A but she is predicted as type B ? Is there any type that is the safest (even if a person is misclassified, there is no life risk)? The most dangerous (there is a life risk)?
According to the answers, you will be able to choose the best metric.
Assuming there is the safest and the most dangerous type, the recall in the dangerous one would be more relevant, because you must identify all of them (maximum as you can).