When you have come face to face with the beast, your idea of minimum protection tends to rise a few levels. If you use the most you can still properly control, something mechanically quick and reliable with quick action sighting, you can't go wrong.
Hunting bear and defending against them are two different things. A .22LR can cause a bear to die, as can the .223. Most people I know start at 30-06 and go up from there, the .338 and 375H&H being quite popular, have been for a long time but there are newer cartridges out now. It doesn't hurt to have a little extra to give you to allow for that margin of error, you want to be sure as a responsible hunter and/or when your life is on the line. You want to have it and not need it than need it and not have it. Your individual skill will play a factor in how much firepower you need to hunt them. If you got the balls, a revolver or bow are sometimes employed (for Blackbear at least) .