One of the historical criteria often invoked by New Testament scholars is the criteria of embarrassment (which is related to the criteria of dissimilarity). It is also called the criteria of divergent patterns, which I think is a more accurate description of the criteria, and the criteria of contradiction. It would encompass not only "embarrassing" information but also historical acts or sayings which do not well serve the early Christian or author's overriding theme. Here is how Darrel Bock describes the criteria:
The criterion of divergent patterns argues that a story that is retained in the face of difficulty is likely to be authentic. For example, texts that are embarrassing to the disciples because of their inadequate responses are more likely to go back to Jesus and are unlikely to have been created by the early church. Sometimes this criterion is called the criterion of embarrassment. Jesus saying he does not know the time of the return in Mark 13:32 is another example. The logic of this criterion is clear: the difficult text was preserved because it belonged in the tradition.
Darrel Bock, Studying the Historical Jesus, page 201.
Perhaps the most well known recent articulation of the criteria comes from John P. Meier:
The point of the criterion is that the early church would hardly have gone out of its way to create material that only embarrassed its creator or weakened its position in arguments with opponents. Rather, embarrassing material coming from Jesus would naturally be either suppressed or softened in later stages of the Gospel tradition, and often such progressive suppression or softening can be traced through the Four Gospels.
Meier, A Marginal Jew, Vol. I, page 168.
Most recently, Craig Keener noted that the criteria of embarrassment is useful only "in its positive role." Keener, The Historical Jesus of the Gospels, page 156. By this he means that the presence of divergent patterns is a factor supporting authenticity but that its absence is not a factor supporting inauthenticity. After all, notable historical figures who launched religious or political movements necessarily said and did things that did not embarrass or contradict their followers or those whom they influenced.
Although it would not seem unusual for specialty fields to adapt or develop their own standards or criteria, it is worth asking whether this criteria or something similar is used by Non-Biblical Historians. According to Robert Stein, historians generally consider material encompassed by the criteria of embarrassment to be valuable.
The inclusion by the Evangelist of material that does not fit his theological scheme serves therefore as a non-intentional witness to the antiquity and authenticity of such material.... This 'despite the author' kind of evidence is furthermore the very kind of evidence that historians often find most valuable.
Robert Stein, "The 'Criteria' for Authenticity," Gospel Perspectives, Vol. I, page 247. Stein cites one of the most influential works of historiography of modern times to support of his statement, Marc Block's The Historian's Craft. I cite a broader portion of Bloch's text:
The narrative sources ... that is, the accounts which are consciously intended to inform their readers, still continue to provide, valuable assistance to the scholar.... Nevertheless, there can be no doubt that, in the course of its development, historical research has gradually been led to place more and more confidence in the second category of evidence, in the evidence of witnesses in spite of themselves.
Marc Bloch, The Historian's Craft, page 61.
Having read the fuller chapter of Bloch, I believe his meaning is broader than Stein's and goes beyond the criteria of embarrassment. However, it is likely broad enough to include the criteria of embarrassment or divergent patterns. In any event, I recently ran across two helpful, recent examples of classical historians using something akin to the criteria of embarrassment.
First, Martin Goodman notes that the most reliable information to be gleaned from Josephus is that which diverges from his own explanations for events.
To accept Josephus' often tendentious evaluation of the motives and characters of the Jews and Romans whose actions constitute his narrative would be rash, but to accept the details of his narrative, particularly when they contradict his own explanations of events, and so survive in the narrative only because they happened, is reasonable.
Martin Goodman, Rome and Jerusalem: The Clash of Ancient Civilizations, page 6.
Next, is Christopher Pelling. I recently cited him as an example of a classicist harmonizing two apparently contradictory ancient accounts. One tool Pelling employs to find one of the accounts likely authentic is that it does not "match" the ancient author's purpose well.
It is surely impossible to believe that [Apollodorus] has himself made up his decree: if he had, he would have made a better job of it, and there would not have been all those mismatches....
Christopher Pelling, Literary Texts and the Greek Historian, page 74-77.
These two examples are useful for more than just showing the employment of "divergent patterns" by Non-Biblical Historians, but also to emphasize that the term "embarrassment" should not be taken too directly. Appollodorus was not "embarrassed" by his account of the Athenians voting to grant Plataean refugees citizenship. He was not involved and it is not going to damage his reputation or embarrass him personally. But, the account does not fit as well as we would expect into the author's overriding theme.
So too when referring to Josephus. It is not that Josephus might be red raced over accounts in his narrative that contradict his narrative asides or conclusory explanations, it is that features of his narratives that diverge from those Josephan characterizations are more likely to come from established sources than Josephus himself. An author is unlikely to creates problems for his overriding purpose if given a choice in the matter. As this criteria recognizes, however, sometimes an author is stuck with the information provided by his sources and not all of them are going to fit well into the argument he is presenting to his audience.
This criteria is one of many, obviously, and it is not a "rule" in that authenticity necessarily follows. It merely is a factor supporting authenticity. It is theoretically possible that the author thought the divergent pattern was an authentic part of the tradition but his source was mistaken. It is also possible that we may be mistaken about what would have been embarrassing. But both Biblical and Non-Biblical Historians alike continue to use this as one tool of historical methodology and with good reason.
Edited to Add: Now for a Non-Biblical Historian using the criteria of embarrassment for a Biblical issue. Michael Grant on John the Baptist' baptism of Jesus:
But the forgiveness of Jesus' own sins, when he was baptized by John, has set the theologians of subsequent centuries a conundrum. For how could Jesus have been baptized for the forgiveness of his own sins when, according to the Christology which developed after his death, he was divine and therefore sinless?
The embarrassment caused by this dilemma is enough to refute modern denials that the Baptist ever baptized Jesus at all. For, once again, the evangelists would have been only too glad to omit this perplexing event; but they could not.
Grant, An Historian's Review of the Gospels, page 49.