I realize that LW collectively doesn't like unreferenced definitions, but in this case maybe it's OK... a friend of mine whose PhD is in decision theory explained aleatory uncertainty to me as the uncertainty of chance with known parameters: if you roll a normal six-sided die, you know it's going to come up with a value in the range 1-6, but you don't know what it will be. There's no chance it will come up 7. Epistemic uncertainty is the uncertainty of chance with unknown parameters: there may not be enough data to know the bounds of an event, or it may have such large and random bounds that trying to place them is not very meaningful.