Link between forward and inverse regression ($\text{E}(X|Y)$ and $\text{E}(Y|X)$ ;$ \text{var}[\text{E}(X|Y)]$ and $\text{var}[\text{E}(Y|X)]$)

Question

In a multivariate context, that is with at least X or Y being a random vector, are there formulae or theorems that link (even remotely) the forward and inverse regression, $\text{E}(X|Y)$ and $\text{E}(Y|X)$ ? Or alternatively $\text{E}(X_i|Y)$ and $\text{E}(Y|X_i)$, where $X_i$ is a given component of $X$.

Similarly, can something be said about $\text{var}[\text{E}(X|Y)]$ and $\text{var}[\text{E}(Y|X)]$ ? Or alternatively $\text{var}[\text{E}(X_i|Y)]$ and $\text{var}[\text{E}(Y|X_i)]$, where $X_i$ is a given component of $X$.

Both $X$ and $Y$ are supposed to be random. The typical application case is that $X$ follows a chosen distribution (for instance a multigaussian) and $Y$ is a function of $X$. The link function is too complicated to be used or even unknown (it is typically a numerical model). The distribution of $X$, however, is controlled. Thus, one should formulate hypothesis on $X$ if needed.

*Note : this question is related to dimension reduction trhough the sliced inverse regression technique.

It is also related to sensitivity analysis, as $\text{var}[\text{E}(Y|X)]$ is the numerator of Sobol' index (surprisingly, the tag sensitivity-analysis does not exist -- if someone with sufficient reputation read this, he might want to create it).*

Answer 1

Note that $\mathbb{E}(X \vert Y)$ heavily depends on this joint distribution, which is not determinated by $\mathbb{E}(Y \vert X)$. Consider the simple case where $X$ and $Y$ are scalar r.vs and $Y=X+\varepsilon$ where $\varepsilon$ has mean zero and is independent of $X$. Depending on the distribution of $\varepsilon$, one can find examples where the function $\mathbb{E}(X \vert Y=y)$ is respectively increasing, constant or decreasing in $y$ for large $y$. Yet in all cases, $\mathbb{E}(Y \vert X=x) \equiv x$.