I bet it's safe to say the 30-06 has been used successfully to kill more brown bears in Alaska (and probably Canada too) then most of the others combined. If I were to use one I'd load with 220 grain bullets and preferable Nosler Partitions.
So because the OP asked what is minimum I would say the 30-06 or ballistic combination in the same range. 7.62X54R, 7.5 Swiss, 8X57 and a handful of others too.
I like my big rifles and I shoot them very well, so I would take one of my more powerful calibers, but if I was to find myself in Alaska again and had one of my 30-06s, (or my 8X57) I would not feel under armed if I got to choose my own ammo. Not at all.
