I've been reading an economics paper regarding rational inattention by Sims (link: https://www.sciencedirect.com/science/article/abs/pii/S0304393203000291) and have been trying to follow his steps in solving the problem and also change the optimization condition regarding the prior.
More generally, I'm trying to solve this specific integral equation:
$$\int_{-\infty}^{\infty}q(y|x)\pi (x)e^{\alpha (y-x)^{2}}dy=1$$
where $$\pi (x) = e^{-\frac{1}{2\sigma ^{2}}(x^{2}+\epsilon x^{4})}$$ and $q(y|x)$ is the unknown posterior probability distribution I'm looking for.
My attempt
If I'm able to move $\pi(x)$ to the right hand side, this equation looks very much like it could be solved with Fourier transforms since the exponential term looks very much like a kernel and could be convoluted with $q(y|x)$. But I'm not sure if that is possible since I don't think I can just divide by $\pi(x)$ on both sides. I don't have the mathematical basis for why this feels wrong, but also $\frac{1}{\pi(x)}$ would not be square integrable.
Where do I go from here?
Well, since $\pi(x)$ can be pulled out of the integration, an attempt to rewrite the expression as a convolution can be made, but it fails thusly:
$$\int_{-\infty}^{\infty}q(y|x)\pi (x)e^{\alpha (y-x)^{2}}dy$$
$$\pi(x) \int_{-\infty}^{\infty}q(y|x)e^{\alpha (x-y)^{2}}dy$$
From which, one may be tempted to write $$\pi(x) \left[q(x|x) * e^{\alpha x^{2}} \right] $$ but that is not correct, as can be seen when one tries to write that convolution operation back into integral form.
I don't have a real answer for you on where to go next.
If you want to pursue writing it as a convolution and using Fourier Transforms, then you need to take steps to rewrite $q(y|x)$ in the integral to get rid of its dependence on $x$.