# Inverse path sampling in (bidirectional) path tracing

In Section 4.1 of the paper Fusing State Spaces for Markov Chain Monte Carlo Rendering, it is described that the CDF $$P$$ of the distribution of paths of length $$k\in\mathbb N$$ may be factored into a product of conditional CDFs, $$P(x_0,\ldots,x_k)=P_0(x_0)\prod_{i=1}^kP_i(x_i\mid x_{i-1},\ldots,x_0).\tag1$$ This is reasonable, since in BDPT a path is incrementally constructed by choosing $$x_i$$ in dependence of its predecessors.

Now they write that "the multivariate variant of inverse transform sampling" generates a path $$(x_0,\ldots,x_k)$$ "by a sequence of inverse transform sampling": \begin{align}x_0&=P_0^{-1}(u_0)\\x_1&=P_1^{-1}(u_1\mid x_0)\\&\cdots\\x_k&=P_k^{-1}(u_k\mid x_{k-1},\ldots,x_0),\tag2\end{align} where $$u_0,\ldots,u_k\in(0,1)$$.

That doesn't make sense to me. They seem to assume that $$x_i$$ is chosen, depending on $$x_0,\ldots,x_{i-1}$$, by drawing a single random number $$u_i\in(0,1)$$. But this is usually not the case$$^\ast$$. Usually, $$x_i$$ is chosen by drawing multiple random numbers (depending on the material of the surface at $$x_i$$).

So, am I missing something? If not, the described "inverse path sampler" doesn't work. Can we fix this? How would we implement it then? I would like to find an implementation in PBRT.

$$^\ast$$ And I think it would be quite weird if the 3-dimensional quantity $$x_i$$ would be chosen by a 1-dimensional variate $$u_i$$.