I don't hang with polar bears and I'm usually promoting spray and noise for grizzlies and just 'tude for black bears, but when I've carried as back-up for grizzlies, a 12-guage auto loaded with slug rounds felt like a good balance - accurate enough at range (50-100 yards) although at distance I'd only be trying to finish a bear off, open sights for close situations, and rapid fire.
Grizzlies eat fish, moose and berries, none of which are very smart.
Polar bears stalk seals, which themselves are predators. Their intelligence and strategy is on a different level.
National Parks in Zimbabwe require a minimal of .375 for walking guides although .458s are common.