Let's say that I have a random variable $X$ distributed according to some prior distribution $p(x)$, for simplicity assume that it's a Log Normal distribution. I then sample $N$ elements from this distribution, which we call $X_i$. Next, let's assume that these samples are weighed according to some importance kernel, such that to each sample there is now associated a weight $\omega_i$ which together represent a posterior (I'm doing SMC).
My problem is the following: I wish to sample from a kernel density representation of the posterior as defined by the weights. But since the distribution is defined on the positive line I'd rather sample from the transformed variable $Y = \log X$ (which is defined on the entire real line). However, I'm not really sure if I can just transform the original, constrained samples $X_i$ to unconstrained samples $Y_i$, fit a KDE from which I draw new samples $\tilde{Y}_i$, and then transform the KDE samples back to the original space via $\tilde{X}_i = \exp{\tilde{Y}_i}$.
While intuitive, this procedure simply feels wrong. Must I e.g. scale the weights by the Jacobian of the transform?
