Originally Posted by BatLobsterRises
Bruce is the one who jumps the gun and assumes the child is Bane. Alfred's intel had him being born in a prison, so when Bruce hears a story being told about a child born there it's only natural to assume the story was about Bane. Bruce didn't allow for the possibility of another child being born there.
The way that Alfred tells the story it comes off as a known legend surrounding Bane, even though it wasn't true. Talia seemed to be mostly unknown outside of the league and the pit prisoners didn't seem to equate the mercenary with Ra's if they even knew who he was. One of the things I love most about the movie is how it deals directly with legends, and not just with Batman.