In the Multiplexed Metropolis Light Transport implementation of the book Physically Based Rendering, the proposal samples are generated by the Primary Sample Space Sampler
In each Markov chain iteration, a random vector $v$ is generated, using the
MLTSampler class, from which the proposed strategy $j$ and a path $y$ are formed. $j$ depends only on the first component of $v$, while $y$ only depends on the remaining entries $v'$ of $v$.
Say $k$ is any other possible strategy. Are we somehow able to obtain the random vector $w'$ which would have produced the same path $y$?
Beyond that I'd need to evaluate the density $q(u,v)$ of the underlying proposal kernel, where $u$ is the random vector from the previous iteration. Is there any way to do that?