I have a probability distribution $p(Z | X)$ from which I can easily sample and compute the probability at every value for $Z$ and $X$. The inverse distribution $p(X | Z)$ however can be very complex and evaluation at given $Z$ and $X$ can be intractable.
I need to calculate the ratio:
$\frac{p(Z | X)}{p(X | Z)}$
It can be seen as a measurement for irreversibility or the increase in entropy.
Is it possible to calculate this ratio without the need to calculate or model the inverse probability distribution ?
Is there literature dealing with similar problems ?
Since $P(X \cap Z)=P(X \mid Z)P(Z) =P(Z \mid X)P(X)$ then providing none of these are zero, you can say $$\frac{ P(Z\mid X ) }{P(X \mid Z)} = \frac{P(Z )}{P(X )}$$ and $${P(X \mid Z)= \frac{ P(Z\mid X ) P(X )}{P(Z )}}$$