While in Yellowstone, I had the pleasure of seeing a Grizzly bear and her two cubs near a park road. The bears did not seem at all threatened and ambled through the bushes, grazing on berries. The raw power of the mother Grizzly seemed more elegant than frightening. Of course, not all bear encounters in the park are this positive. You can read about a deadly bear attack here.
When fishing in the Yellowstone backcountry or anywhere else that Grizzly bears frequent, it is vital to keep a can of bear spray at the ready. Bear spray is pepper mace that can be sprayed at a Grizzly to deter an attack. A study by a BYU professor, posted here, shows that bear spray is more effective than a gun in fending off a charge. The spray should not be sitting in your backpack, not still wrapped in plastic, not tucked away in your fishing vest, but on the belt or on the chest, ready to be used. Otherwise, why take it?
A second point to remember is that bears hate surprises. When hiking alone, be sure to call out "hey bear" at every blind turn and make a lot of noise to alert bears to your presence. When in a group, keep up a steady conversation and again, shout "hey bear," at blind turns on the trail.
A third point is to safeguard your food. Don't leave a pack full of food on the river bank while you are fishing. Take the proper precautions in camp to keep food away from Yogi. Don't camp next to where you cooked a fresh caught trout or even top-ramen. Hang food on trees or keep it in a "bear can," but be sure to consult park rangers to see which method they recommend.
While you should take these safety measures, it is important to remember that buffalo charges and vehicle accidents cause more deaths in Yellowstone than bear attacks. On my trip, I saw a sleepy driver slam into a tree and a small RV catch on fire. Sometimes our fears can magnify certain risks and minimize others.