Cognitive biases play a huge role in The Crucible. Broadly defined, one who engages in cognitive bias has a skewed, irrational, and/or generally incorrect perception about something in the world. Many of the authority figures in the play engage in cognitive bias for various reasons. Basically, there is a process from perceiving objective evidence in the world and this process is manipulated by their subjective frames of thinking. In other words, someone with a cognitive bias takes a rational, objective event and in the process of perceiving it, he/she changes it, bends it, or flat out contradicts it.
The most prevalent example of cognitive bias in The Crucible is confirmation bias. This is when one looks for things to confirm what one already believes, or wants to believe. Danforth is completely sure that the girls are not lying:
I have seen people choked before my eyes by spirits; I have seen them stuck by pins and slashed by daggers. I have until this moment not the slightest reason to suspect that the children may be deceiving me.
Despite the fact that he is supposed to be an objective judge, he has already made up his mind. With that kind of stubborn, single-minded view he is only looking for evidence that will confirm what the girls have been saying. He hears what John has to say but he is skeptical all the time.
Since Danforth, Hathorne, and Parris (the former two representing the "court") have made up their minds, they also are guilty of confirmation bias and self-serving bias. When anyone challenges the girls' claims (that of the presence of witchcraft), that one is challenging the court. Danforth makes this clear. Danforth also commits "self-serving bias" on his own. When Hale (who overcomes any cognitive bias he may have had) confronts Danforth, asking to at least postpone the executions, Danforth disagrees, saying that it would make his (and the court's) authority look bad. He is now engaging in willful cognitive and self-serving bias, a complete denial of the now very real possibility (with Hale's expertise) that all those convicted are innocent.