Let $Y_1$ and $Y_2$ be independent r.v.s, and $X$ be another random variable.
What is $E[X|Y_1, Y_2]$? Is $E[X|Y_1, Y_2]$ equivalent to $E[X|Y_1] \cdot E[X|Y_2]$?
More specifically, is there a formula for the density of $f_{X|Y_1,Y_2}$ just like $f_{X|Y_1}$$=$$f_{X,Y_1}/f_{Y_1}$ (assuming that regular conditions such as $f_{Y_1}>0$ are satisfied)?
What is $E[X|Y_1,Y_2]$?
Literally, the expectation of $X$ given the values of both $Y_1$ and $Y_2$.
"The average height (X) of a 12 year old (Y1) male (Y2)" is an example.
Is $E[X|Y_1,Y_2]$ equivalent to $E[X|Y_1]⋅E[X|Y_2]$?
No. Consider the trivial case where $X$ is independent of both $Y_1$ and $Y_2$. You'd be asking "Is $E[X]$ equivalent to $E[X]⋅E[X]\,$?"
Or in the "average height of a 12 year old male" case, it's asking "is that the same as the average height of a 12 year old times the average height of a male?" ... which is plainly not the case.
is there a formula for the density of fX|Y1,Y2 just like fX|Y1=fX,Y1/fY1 (assuming that regular conditions such as fY1>0 are satisfied)
Of course!
$f_{X|Y_1,Y_2}=f_{X,Y_1,Y_2}/f_{Y_1,Y_2}$
