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

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    $\begingroup$ These expressions relate to Bayes rule. What you are trying to infer is the posterior distribution without a prior distribution. That is not possible. $\endgroup$ Commented Mar 19, 2022 at 14:21

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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 $p(x\vert y)$.

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    $\begingroup$ which can be obtained via the joint $\endgroup$
    – gunes
    Commented Mar 19, 2022 at 11:05
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    $\begingroup$ @gunes Is that joint distribution known? The question statement tells that we only know $p(y|x)$ and p(y)$. $\endgroup$ Commented Mar 19, 2022 at 14:22
  • $\begingroup$ seems like I misread it as p(x|y) $\endgroup$
    – gunes
    Commented Mar 19, 2022 at 14:27

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