I am studying SPA for LDPC decoding. This is one of the PDFs I am studying and my question is on page 12 of it. http://tuk88.free.fr/LDPC/ldpcchap.pdf
In binary-input AWGN channel, $x_i=\pm 1$ when $c_i$ is $0,1$ then the received signal $y_i=x_i+n_i$.
All the books I've read say it is easy to show that
$$Pr(x_i=x | y_i)=[1+exp(-2yx/\sigma^2)]^{-1}$$
and use the formula when calculating APP(a posterior probability).
But I don't know how this formula is derived. Can you give me an idea of how this was induced?