Multiple Importance Sampling in Bidirectional Path Tracing

Question

I'm trying to understand the computation of the path density described in the book Physically Based Rendering. They assume that a path $$x=(x_0,\ldots,x_{n-1})=(q_0,\ldots,q_{s-1},p_{t-1},\ldots,p_0)$$ with $s$ and $t$ vertices on the light and camera path, respectively, has been generated. Along with each vertex $x_i$ they store two densites $\overset{{}_{\rightarrow}} p(x_i)$ and $\overset{{}_{\leftarrow}} p(x_i)$. $\overset{{}_{\rightarrow}} p(x_i)$ is the "forward" density of $x_i$, which is probability per unit area of $x_i$ as generated by the path sampling algorithm. $\overset{{}_{\leftarrow}} p(x_i)$ is the hypothetical probability density of $x_i$ if the direction of light transport was reversed.

Now let $p_i(x)$ denote the density of sampling the same path $x$ with the strategy $(i, j)$, $i + j = s + t$, instead (i.e. the strategy of considering $x_0,\ldots,x_{i-1}$ and $x_i,\ldots,x_{n-1}$ as being on the light and camera path, respectively). They claim that $$p_i(x)=\overset{{}_{\rightarrow}} p(x_0)\cdots\overset{{}_{\rightarrow}} p(x_{i-1})\overset{{}_{\leftarrow}} p(x_i)\cdots\overset{{}_{\leftarrow}} p(x_{n-1})\tag1.$$

I can't make sense of $(1)$. Even if we take $i=s$ (in which case the $(i, j) = (s, t)$ is the strategy being used for generating $x$), the formula doesn't make sense to me. Shouldn't $p_s(x)$ be the product of the densities of the light and camera subpaths? The density of the light subpath is clearly $\overset{{}_{\rightarrow}} p(x_0)\cdots\overset{{}_{\rightarrow}} p(x_{i-1})$, since the sampling startet at a light source at $x_0$ and all the densities on the way to $x_{i-1}$ are "forward densities". On the other hand, the density of the camera subpath should be $\overset{{}_{\rightarrow}} p(p_0)\cdots\overset{{}_{\rightarrow}} p(p_{t-1})$, since from $x_{n-1}=p_0$ all the densities are forward densities as well. However, according to $(1)$ the density of the camera subpath is $\overset{{}_{\leftarrow}} p(p_{t-1})\cdots\overset{{}_{\leftarrow}} p(p_0)$. What am I missing?