Modest Epistemology is the claim that average opinions are more accurate than individual opinions, and individuals should take advantage of this by moving toward average opinions, even in cases where they have strong arguments for their own views and against more typical views. (Another name for this concept is "the wisdom of crowds" -- that name is much more popular outside of LessWrong.) In terms of inside view vs outside view, we can describe modest epistemology as the belief that inside views are quite fallible and outside views much more robust; therefore, we should weigh outside-view considerations much more heavily.
In LessWrong parlance, "modesty" and "humility" should not be confused. While Eliezer lists "humility" as a virtue, he provides many arguments against modesty (most extensively, in the book Inadequate Equilibria; but also in many earlier sources.) Humility is the general idea that you should expect to be fallible. Modest Epistemology is specifically the view that, due to your own fallibility, you should rely heavily on outside-view. Modest epistemology says that you should trust average opinions more than your own opinion, even when you have strong arguments for your own views and against more typical views.
Historically, Robin Hanson has argued in favor of epistemic modesty and outside-view, while Eliezer has argued against epistemic modesty and for a strong inside views. For example, this disagreement played a role in The Foom Debate. Eliezer and Hanson both agree that Aumann's Agreement Theorem implies that rational agents should converge to agreement; however, they have very different opinions about whether/how this breaks down in the absence of perfect rationality. Eliezer sees little reason to move one's opinion toward that of an irrational person's. Hanson thinks irrational agents still benefit from moving their opinions toward each other. One of Hanson's arguments involves pre-priors.
External Posts:
Immodest Caplan by Robin Hanson
Related Sequences: Inadequate Equilibria
Related Pages: modesty, Humility, Inside/Outside View, Egalitarianism, Modesty argument, Disagreement