This is a sort of paradox that has me confused. i just need some help clearing things up.
Let $x,y$ be two discrete, finite variables, where $x$ can take on $N$ values and $y$ can take $M$ values. Suppose someone hands you a table of the numbers $P(x|y)$. Can you determine $P(x,y)$? Intuitively, I think the answer is No. However, consider the following system of equations:
$$P(x|y)\sum_\xi P(\xi,y) = P(x,y)$$
(This is just another way to write the product rule $P(x|y)P(y)=P(x,y)$.)
Here you have $M\times N$ linear equations in the $M\times N$ unknowns $P(x,y)$. So it seems that $P(x,y)$ could be determined from this system of equations. If this is true, then knowledge of the $M\times N$ numbers $P(x|y)$ are enough to determine the $M\times N$ numbers $P(x,y)$, but this doesn't feel right. What am I doing wrong?