I thought that "self defense" was something you had to prove yourself, where you've admitted to committing a crime (killing someone) but that you had a specific

I thought that "self defense" was something you had to prove yourself, where you've admitted to committing a crime (killing someone) but that you had a specific circumstance that mitigated the charge. I assumed it was similar to a "not guilty by reason of insanity" plea, where you admit you committed a crime but there are special circumstances (that you have to prove) that mitigate it. You can't just stay silent the entire time and force the prosecutor to prove you were insane - you have to actually provide some evidence you were insane. Is this not true?