Everyone is assuming that you can't identify the person on the other side of the door, and saying that shooting through a door is highly irresponsible. I'd like to point out that a lot of doors, especially a lot of back doors and side doors, have large window panes in them, and therefore it's possible you could identify the target and what is beyond it.
That being said, unless the person is clearly trying to break in or is armed and appears to have ill intent, shooting through the door is still a bad idea from a legal and moral standpoint.
If someone was banging on my door at 4am, I'd prepare for ill intent (get my gun and have my wife with phone in hand ready to call 911), but before calling the police, I'd check to find out what the situation was. That includes going to a location where I can see what's going on, and verbally challenging the person at the door to find out who they are and what they wanted.
However, I don't think you'd want to stand close to and in line with the door (in case they were armed and had no compunction against firing through a door), and I don't think you'd want to illuminate yourself by turning the lights on until you had a better understanding of whether the individual outside might have intent to harm you.
I also prefer not to advertise that I have a weapon. While that might discourage the average bad guy, some might just alter their tactics to be prepared to deal with an armed person. I would instead say that the police have been called (whether we'd called them yet or not).
I don't believe in calling the police until I've determined that there is a need. Too much of that can lead to the situation of Peter and the Wolf.