I can't reproduce the derivation of the "stretch" move in Goodman and Weare (2010) and it's bugging me. I have two questions:
1) How is this proposal symmetric? (cf. eq. 7) $$X_k(t) \to Y = X_j + Z(X_k(t)-X_j)$$ where $X_k(t)$ is the position of a walker $k$ at step $t$, $Y$ is the proposal for $t+1$, $X_j$ the pos. of another walker and $Z$ is a random variable with density $g(z)$
Given a random variable $z$ with density $g(z)$, for $Pr(X_k\to Y)=Pr(Y\to Y_k)$ (cf. eq. 8), wouldn't $g$ have to satisfy $g(z^{-1})=g(z)$, not $g(z^{-1})=z~g(z)$? Where did the $z$ come from?
2) How is the conditional distribution of $\pi$ along the ray $$S=\{Y\in\mathbb{R}^n:Y-X_j=\lambda(X_k-X_j),\lambda>0\}$$ derived as $f(Y)\propto\|Y-X_j\|^{n-1}\pi(Y)$, which at the very least implies that when restricted to $S$ the likelihood of a given $Y$ increases with distance from $X_j$ even though this is contrary to the chosen density $g(\lambda)$. There's also the fact that this isn't a legitimate density if $\pi$ vanishes slower than the leading factor.
EDIT: I think a good candidate for a symmetric likelihood is $\propto\phi(\log z)$ where $\phi$ is a normal kernel with zero mean and variance equal to $\sigma^2$. This can be simulated easily by taking the exponential of a normal random variable with variance equal to the mean, the parameter $a$.