It's pretty simple. What are your priorities? If killing a caribou is the absolute top priority you should go equipped to accomplish that.
If your priority centers around killing one with your bow, go into it with a bow and plenty of determination.
If you don't bring a gun there's a 100% chance that any caribou you kill will be by using a bow.
If you bring both weapons, my personal estimate is you've probably got an 80% chance your kill will be while using the gun.
I'm biased toward using a bow for my own hunting. I've never been the guy who just HAD to kill an animal to call it a great hunt. I've never had the pull to kill a big game animal with a long range weapon such as scoped bolt rifle. Every big game hunt I've ever done, I rolled the dice willingly by going into it with only a bow. The times I didn't score probably only served to make me more determined to get better and hunt harder with a bow. Some of us are just hardcore bow people. That might not be a great thing for success rates, but
how I hunt it has always been more important than
if I kill it.
The bottom line is to know yourself and what makes you happy. Don't let my choices and values affect your choices and values. I don't really care what the other guy does....it's his hunt.