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Say we have fixed a length $k$ for paths we want to sample using bidirectional path tracing as described in the Physically Based Rendering book. Let $q_{s,\:t}$ denote the probability density of sampling a path with $s$ vertices on the light and $t$ vertices on the camera/eye subpath ($s+t-1=k$).

Is there a relation between the $k+2$ different densities (obtained by varying $s=0,\ldots,k+1$)? In particular, given a path $x$ and $(s,t)$, can we factorize $q_{s,\:t}(x)=q_{s_1,\:t_1}(y)q_{s_2,\:t_2}(z)$ for suitable $s_1,t_1,s_2,t_2,y,z$?

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