Infer $p(x|y)$ given $p(y|x)$ and $p(y)$ Given $p(y)$ and $p(y|x)$, how can we infer $p(x|y)$?
Or to what extent can we know about $p(x, y)$?

By using $p(x,y)$ as the middle ground, we can obtain
$$
\begin{aligned}
p(x\vert y) p(y) &= p(y\vert x)p(x)\\
p(x\vert y) &= \frac{p(y\vert x)}{p(y)}p(x).
\end{aligned}
$$
Therefore knowledge about $p(x)$ is required for us to obtain $p(x\vert y)$.
