I am trying to solve this integral:
$$t(v)\equiv\sum_{k=1}^{n}\sum_{j=1}^{n}\int_{-\infty}^{\infty}\frac{w_{k}N(x-x_{k},B_{k})N(x-x_{j},v)}{\sum_{m=1}^{n}w_{m}N(x-x_{m},B_{m}+v)}dx $$
where $\sum_{k=1}^{n}w_{k}=1$, $N(x,q)\equiv\frac{1}{\sqrt{2\pi q}}e^{-\frac{x^{2}}{2q}}$, {$v,B_1,...B_n,w_1,...w_n,$} are all real and positive, and {$x_1,...x_n$} are all real.
This integral can also be written as
$$t(v)\equiv\sum_{j=1}^{n}\int_{-\infty}^{\infty}\frac{f(x)N(x-x_{j},v)}{p(x,v)}dx$$
where $f(x)\equiv\sum_{k=1}^{n}w_{k}N(x-x_{k},B_{k})$ and $p(x,v)\equiv f(x)*N(x,v)$
Discussion:
1) The only way I could think of to solve or partially solve this is with Residue methods, but I welcome any method that works.
2) I'm not certain if the roots of a Gaussian Mixture of $n>2$ terms can be expressed analytically.
3) Given a root of $p[x,v]$, I'm not certain how to calculate the residue of the corresponding pole.
4) Given the infinite roots of $p[x,v]$ and the corresponding residues of the integrand, I'm not certain how to select a path of integration that will solve this problem.
5) Somewhere, at some point in my life, I have seen some problems of the form $\int_{-\infty}^{\infty}\frac{g(x)}{h(x)}dx$ where if certain relationships exist between $h(x)$ and $g(x)$, it was possible to solve the integral via residue-related methods without having to explicitly find the roots of $h(x)$. Can those methods be applied here? Can anyone point me to a reference to such methods if they recall them?
I suspect the result might come out involving $\sum_{j,k}\frac{N(x_{j}-x_{k},v)}{p(x_{k},v)}$ but I am not certain.
Thanks in advance for any help or insights.
