The criterion of embarrassment is a form of analysis in which an account embarrassing to the author is presumed to be true because the author would have no reason to create an embarrassing account about themself.
Norman L. Geisler and Frank Turek explain how this relates to the New Testament:
One of the ways historians can tell whether an author is telling the truth is to test what he says by “the principle of embarrassment.” This principle assumes that any details embarrassing to the author are probably true. Why? Because the tendency of most authors is to leave out anything that makes them look bad. How does the New Testament measure up to the principle of embarrassment? Let’s put it this way: If you and your friends were concocting a story that you wanted to pass off as the truth, would you make yourselves look like dim-witted, uncaring, rebuked, doubting cowards? Of course not. But that’s exactly what we find in the New Testament.
If you were a New Testament writer, would you include these embarrassing details if you were making up a story? Would you write that one of your primary leaders was called “Satan” by Jesus, denied the Lord three times, hid during the crucifixion, and was later corrected on a theological issue? Would you depict yourselves as uncaring, bumbling cowards, and the women—whose testimony was not even admissible in court—as the brave ones who stood by Jesus and later discovered the empty tomb? Would you admit that some of you (the eleven remaining disciples) doubted the very Son of God after he had proven himself risen to all of you? Of course not.
In short, we don’t have enough faith to believe that the New Testament writers included all of those embarrassing details in a made-up story. The best explanation is that they were really telling the truth—warts and all.– Norman L. Geisler and Frank Turek – 10 Things You Should Know about the Reliability of the New Testament Writers
Why would the disciples write things that would only embarrass them?