Let $y$ is a Gaussian Random variable, how to get the following result?
$ln[\frac {P(y_1 ,y_2 |x=1)} {P(y_1 ,y_2 |x=-1)}]$ $= ln[\frac{1+exp(v_1 +v_2)}{exp(v_1)+exp(v_2)}]$
Where $v_i = 2y_i/\sigma^2$
$i=1,2$ here and $\sigma^2$ is a variance of Gaussian R.V.