If $X$ and $Y$ are uncorrelated random variables, then under what condition is $E[X \mid Y] \approx E[X]?$ Suppose $X$ and $Y$ are real random variables that are uncorrelated. Now, uncorrelated does not imply independence, so $E[X \mid Y] \ne E[X]$.
However, can they be said to be approximately equal? If so, under what conditions does that approximation hold? (I realize that the equality is exact when $X,Y$ is a multivariate Gaussian, but I want to know more generally when the equality can be approximated.)
As a bonus, is it possible to do an expansion of $E[X \mid Y]$ that looks like "$E[X] + $higher order terms", so we can see explicitly when those higher order terms can be said to be negligible? Let's assume that $p(X \cap Y)$ is continuous and infinitely differentiable.

One situation where this is interestingly almost true is when $X$ and $Y$ are both projections from the same high-dimensional distribution, ie, $X=a^TZ$, $Y=b^TZ$ for high-dimensional $Z$.  Hall & Li (and earlier work by various people) showed that for 'most' distributions for $Z$ on a high-dimensional sphere and most $a,b$, $E[X|Y]$ is approximately linear in Y and so if they are uncorrelated they are close to independent.
The result makes sense, because $X$ and $Y$ will be approximately bivariate Gaussian by the CLT, but actually pinning down the error bounds takes work.
This question was motivated by the sliced inverse regression method of Duan and Li, where you regress $X$ on $Y$ to learn about $Y|X$
