I believe ghosts could exist because I'm trying to keep an open mind. When animals go slightly strange and start chasing things or staring at something we can't see it's kinda creepy, right? Well, one of my cats does it in my room and actually tries to climb up the same corner of my room all the time, and often even starts walking around chasing whatever he sees in the air, looking up at it all the time. I don't know what he's seeing, but one of the many possibilities could be a ghost. Animals can see better than humans, so they might be able to see ghosts whereas sometimes we can only see them when the conditions are right.
Personally, I'd like to believe something my religious studies teacher was talking about: that ghosts and spirits are different things. I'm not saying I believe IN ghosts or spirits, but I'm saying if they do exist then I'd like to believe it in the way he explained it. Basically, the spirit is the actual person and moves on or is reborn or whatever, and the ghost is like a leftover shadow of the person that plays on a continual loop. We can't interact with it because it can't see us, it's like... in a different dimension, or plane or existance, however you want to think of it. But yeah, that's sort of what I'd like to believe, that there's something after death, and that spirits are real and sometimes visit us and ghosts are shadows of them and stuff. This is the thing which makes sense to me, as my cat always follows in the same direction, so for all I know I have some creepy ghost in my room. I don't actually think that, he's probably chasing dust, but you never know.
EDIT: Also, I forgot. There's another reason I believe they might exist in a different way than I mentioned above. At my grandma's house (nice grandma, the psycho one is just ugh), in the room she used to sleep in (she doesn't anymore, she sleeps in the small room) you can hear this breathing if you listen carefully. My mum's heard it for years, so has grandma and my dad's even heard it. I've heard it too because I used to live there and sleep in that room. It's not scary or anything. But my family has a tendency for seeing ghosts and things which aren't there on my dad's side. My mum's side only has that, but my dad's side? Well, they're all crazy but for all I know they really do see ghosts and stuff. They don't like to talk about it, though, so I don't know a lot.