For the past 10+years I've had bear encounters or sightings in the NW every year, while hiking. I've never encountered a bear while biking, but did encounter a cougar while mountain biking in Colorado.

I carry bear spray while hiking and have used it once, against a bear in Colonel Bob Wilderness (Washington) The spray did not do much to the bear, even though it was right in the face. I am convinced it ran away mostly out of fear, which is typical. IMHO spray is not very effective against large black bears, and is probably not at all effective against large grizzlies. But I still carry bear spray when hiking.

Capsaicin (bear) spray is very effective against dogs, and I've used spray against dogs many times while biking. I bike a lot in rural areas and people seem to let the dogs run free. For this reason alone, I recommend taking a can of spray while biking.

If you have to spray a bear while biking, I wish you the best of luck.