I'm in the middle of a proof concerning Bayes's theorem, and ran into a bit of a wall. I came up with the following term:
$\int_{-\infty}^{\infty}p_W(w_2|\theta)[\frac{p_W(w_1|\theta)f_\Theta(\theta)}{\int_{-\infty}^{\infty} p_W(w_1|\theta)f_\Theta(\theta)d\theta}]d\theta$
I'm not sure I'm formatting it correctly, but one of the factors in the integrand of the outer integral contains another integral in its denominator, with respect to the same variable as the outer integral. At this point I have virtually no knowledge of what to do to rearrange terms.
Is the following a possible rearrangement of the above expression, considering the outer integral to be part of the numerator? I'm hoping so because it would make my proof work :)
$\frac{\int_{-\infty}^{\infty}p_W(w_2|\theta)p_W(w_1|\theta)f_\Theta(\theta)d\theta}{\int_{-\infty}^{\infty}p_W(w_1|\theta)f_\Theta(\theta)d\theta}$
If it's not, any suggestions on how these terms might otherwise be rearranged? To be clear, I'm not asking for any simplification with respect to probabilities, or evaluation of integrands, etc., I'm purely asking how these terms can be rearranged. I'm not certain on what I can do with the integrals, specifically.