Given random variables Y, X, and C how do we prove the following: $E[Var(E [Y| X, C ]| C)] = E[(Y-E[Y|X,C])^2] - E[(Y - E[Y|C])^2]$. This property is the crux of the following paper: https://aclanthology.org/N18-1146.pdf which I am trying to understand.
From Wikipedia's article on total variance, I can see the following decomposition: $\operatorname{Var}[Y] = \operatorname{E}[\operatorname{Var}(Y\mid X, C)] + \operatorname{E}[\operatorname{Var}(\operatorname{E}[Y\mid X,C]\mid C)] + \operatorname{Var}(\operatorname{E}[Y\mid C])$, but don't yet see how it is equal to $E[(Y-E[Y|X,C])^2] - E[(Y - E[Y|C])^2]$
By the law of total conditional variance, $$\operatorname{Var}(Y|C) = E[\operatorname{Var}(Y|X,C)|C] + \operatorname{Var}(E[Y|X,C]|C).$$ Apply the expectation and simplify using the law of total expectation: $$E[\operatorname{Var}(Y|C)] = E[\operatorname{Var}(Y|X,C)] + E[\operatorname{Var}(E[Y|X,C]|C)]$$ Now just rearrange: $$E[\operatorname{Var}(E[Y|X,C]|C)] = E[\operatorname{Var}(Y|C)] - E[\operatorname{Var}(Y|X,C)]$$ which differs from what you've written by a sign.