# Factorize probability densities in Bidirectional Path Tracing

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 $$s_1,t_1,s_2,t_2,y,z$$?