pr.probability – Lower bound for reduced variance after conditioning

Let $$X$$ be a random variable with variance $$tau^2$$ and $$Y$$ be another random variable such that $$Y-X$$ is independent of $$X$$ and has mean zero and variance $$sigma^2$$. (One can think of $$Y$$ as a noisy observation of $$X$$.) It follows from the law of total variance that $$mathbb{E}(operatorname{Var}(X|Y))leqoperatorname{Var}(X)$$. Under normality, it is known that $$operatorname{Var}(X|Y)=frac{sigma^2tau^2}{sigma^2+tau^2}=tau^2left(1-frac{tau^2}{sigma^2+tau^2}right)$$ almost surely, and this inequality is stronger than $$mathbb{E}(operatorname{Var}(X|Y))leqoperatorname{Var}(X)$$ as it quantifies how much the expected variance is reduced.

I wonder if one could prove something similar in general. If this does not hold in general, would it help to assume the random variables are sub-Gaussian?