I have $S=R+\epsilon$ where $R \sim Cauchy(r, 1/\alpha)$ and $\epsilon \sim Cauchy(0, 1/\beta)$. I want to calculate the distribution of $R$ given $S=s$.
I've tried the following:
By the definition of conditional probability, we have $f_{R | s}(x | s) = \frac{f_R(x) f_\epsilon(s-x)}{f_{R+\epsilon}(s)}$.
We know that for $X \sim Cauchy (x_0, \gamma)$ we have the density function $f(x) = \frac{1}{\pi \gamma} \frac{\gamma^2}{(x-x_0)^2 + \gamma^2}$
Then plugging this in (and simplifying a bit), we get
$$f_{R | s}(x | s) = \frac{1}{\pi (\frac{1}{\alpha + \beta})} \frac{\frac{1}{(\alpha+\beta)^2}}{\frac{((x-r)^2+\frac{1}{\alpha^2})((s-x)^2+\frac{1}{\beta^2})}{(x-r)^2+(\frac{\alpha+\beta}{\alpha \beta})^2 } } $$
I can't simplify the part on the bottom to get what would be desired if it were a Cauchy distribution. Any hints/tips? Thank you!