For example, some people have an irrational belief in supernatural beings.

Or that they have irrational beliefs about them?

One more reason for irrational beliefs can perhaps be explained by operant conditioning.

They'd developed a blind irrational belief that their imprisonment was unnecessary.

Americans are doomed by their submission to irrational religious beliefs.

Yet, you'll often find rational thinkers using science to prove some incredibly irrational beliefs.

I suggest the correct description is Micawberite - an irrational belief that "something will turn up"

Religion at its most liberal is still an extreme irrational belief.

Most participants show signs of optimism bias (the irrational belief that they will beat the odds).

Evidence of persistent irrational beliefs or negativistic thinking regarding the diagnosis.