In the Multiplexed Metropolis Light Transport implementation of the book Physically Based Rendering, the proposal samples are generated by the Primary Sample Space Sampler MLTSampler.

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?


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