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I'm having trouble combining variables from differents tables.

In this case, I have the exogenous variables $X$ (categorical) and the endogenous variable $Y$ (continuous). None of these variables are independant.

  • Table A has the variables $x_1, x_2, x_3$ and $Y$.
  • Table B has the variables $x_1, x_4$ and $Y$.
  • Table C has the variables $x_1, x_2, x_3, x_4$, whithout any reference to $Y$.

It's easy to compute from the table A $E [ Y | x_1, x_2, x_3 ]$, and from table B $E [ Y | x_1, x_4 ]$. But my main goal is to compute $E [ Y | x_1, x_2, x_3, x_4 ]$ and I can't find a way to do it.

It might help to know that from table C, I can also get the conditional probability $P[x_4 | x_1, x_2, x_3]$ as well as $P[x_2 | x1]$ and $P[x_3 | x1]$.

One of my try was to use the formula $E[Y] = E[E[Y|X]]$ (with all $x_i$) as I am able to know $E[Y]$. But then I would have to solve the following sum $\sum y \cdot P[Y | x_1, x_2, x_3, x_4]$, and I believe I can't build the requiered values for $y$ (or don't know how).

Any idea how to get $E [ Y | x_1, x_2, x_3, x_4 ]$ ?

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