  We introduce a notion of uniform Ding stability for a projective manifold
with big anticanonical class, and prove that the existence of a unique
K\"ahler-Einstein metric on such a manifold implies uniform Ding stability. The
main new techniques are to develop a general theory of Deligne functionals -
and corresponding slope formulas - for singular metrics, and to prove a slope
formula for the Ding functional in the big setting. This extends work of Berman
in the Fano situation, when the anticanonical class is actually ample, and
proves one direction of the analogue of the Yau-Tian-Donaldson conjecture in
this setting. We also speculate about the relevance of uniform Ding stability
and K-stability to moduli in the big setting.
