In reasoning under uncertainty in AI, there are (at least) two useful and different ways of understanding beliefs: the first is as absolute belief or degree of belief in propositions and the second is as belief update or measure of change in belief. Pignistic and plausibility transformations are two well-known probability transformations that map belief functions to probability functions in the Dempster-Shafer theory of evidence. In this paper, we establish the link between pignistic and plausibility transformations by devising a belief-update framework for belief functions where plausibility transformation works on belief update while pignistic transformation operates on absolute belief. In this framework, we define a new belief-update operator connecting the two transformations, and interpret the framework in a belief-function model of parametric statistical inference. As a metaphor, these two transformations projecting the belief-update framework for belief functions to that for probabilities are likened to the fire projecting reality into shadows on the wall in Plato's cave.