Between a polar bear and a grizzly it would probably boil down to size. Although very different in appearance, both species are genetically very close. According to some accounts I've read, their genetic separation occurred only about 100,000 years ago, less than some of the "races" of modern humans. I believe that hybridization between them is possible, but I am not sure...or if the offspring are fertile.
Liger males are sterile, FWIW, but the females are fertile and can be bred back to male lions or tigers.
Between a tiger and a grizzly, my first instinct would be to put my money on the grizzly. I have read that tigers typically avoid black or sloth bears in the wilds of Asia, although these are much less formidable bears than grizzlies. I suppose the Amur (Siberian) tigers must come into contact with brown bears that are for practical purposes identical to our grizzlies, but I've never read about a fight.
HOWEVER, in the old days in California, it was said that a mountain lion could kill a grizzly and did so on occasion, even though the California grizzly was supposed to be a very large race, almost comparable to the coastal brown bears of Alaska. Since a tiger is a vastly more formidable animal than the mountain lion, the implications are obvious.
I think fights like African lions vs. grizzlies and such were staged in places like Mexico at least in the old days. Maybe I'll look into some of these topics.