The "-i" option actually means "simulate Initial login" -- the idea being -i will drop you into a shell as if you'd logged in as the user in a fresh session. -s will also give you an interactive shell, but doesn't mess with the environment.
(And both -i and -s can also not give you an interactive shell, if you give it a command to run. The point is that it will run the command in the shell rather than just fork()ing and exec()ing like a bare sudo invocation would do.)
I'm not objecting to that. I'm objecting to the (perhaps unintended) assertion that "sudo -i" means "give me an interactive shell". It does happen to give you an interactive shell (assuming you don't specify a command), but as a side effect. "-s" gives you an interactive shell too, but the semantics are different.
One side effect of "sudo bash" is that when you exit that shell your local user's .bash_history file gets overwritten by one owned by root (and un{read/write}able by the local user), and so your local user loses the bash history functionality from that point on.