Appendix CPathway Sums for Chain Graphs,${\mathsc{𝒮}}_{\alpha ,\beta }^{{C}_{N}}$

To obtain the total probability of leaving the chain ${C}_{N}$ via node $\alpha$ if started from node $\beta$, i.e${\mathsc{ℰ}}_{\alpha }^{{C}_{N}}{\mathsc{𝒮}}_{\alpha ,\beta }^{{C}_{N}}$, we must calculate the pathway sum ${\mathsc{𝒮}}_{\alpha ,\beta }^{{C}_{N}}$. We start with the case $\alpha =\beta$ and obtain ${\mathsc{𝒮}}_{\beta ,\beta }^{{C}_{N}}$. Consider any path that has reached node $N-1$. The probability factor due to all possible $N-1$ to $N$ recrossings is simply ${R}_{N-1}=1∕\left(1-{P}_{N-1,N}{P}_{N,N-1}\right)$. We need to include this factor every time we reach node $N-1$ during recrossings of $N-2$ to $N-1$. The corresponding sum becomes

 ${R}_{N-2}=\sum _{m=0}^{\infty }{\left({P}_{N-2,N-1}{P}_{N-1,N-2}{R}_{N-1}\right)}^{m}=\frac{1}{1-{P}_{N-2,N-1}{P}_{N-1,N-2}{R}_{N-1}}.$ (C.1)

Similarly, we can continue summing contributions in this way until we have recrossings of $\beta$ to $\beta +1$, for which the result of the nested summations is ${R}_{\beta }=1∕\left(1-{P}_{\beta ,\beta +1}{P}_{\beta +1,\beta }{R}_{\beta +1}\right)$. Hence, ${R}_{\beta }$ is the total transition probability for pathways that return to node $\beta$ and are confined to nodes with index greater than $\beta$ without escape from ${C}_{N}$.

We can similarly calculate the total probability for pathways returning to $\beta$ and confined to nodes with indices smaller than $\beta$. The total probability factor for recrossings between nodes 1 and 2 is ${L}_{2}=1∕\left(1-{P}_{1,2}{P}_{2,1}\right)$. Hence, the required probability for recrossings between nodes 2 and 3 including arbitrary recrossings between 1 and 2 is ${L}_{3}=1∕\left(1-{P}_{2,3}{P}_{3,2}{L}_{2}\right)$. Continuing up to recrossings between nodes $\beta -1$ and $\beta$ we obtain the total return probability for pathways restricted to this side of $\beta$ as ${L}_{\beta }=1∕\left(1-{P}_{\beta -1,\beta }{P}_{\beta ,\beta -1}{L}_{\beta -1}\right)$. The general recursive definitions of ${L}_{j}$ and ${R}_{j}$ are:

 (C.2)

We can now calculate ${\mathsc{𝒮}}_{\beta ,\beta }^{{C}_{N}}$ as

 (C.3)

where we have used Equation C.2 and the multinomial theorem [242].

We can now derive ${\mathsc{𝒮}}_{\alpha ,\beta }^{{C}_{N}}$ as follows. If $\alpha >\beta$ we can write

 ${\mathsc{𝒮}}_{\alpha ,\beta }^{{C}_{N}}={\mathsc{𝒮}}_{\alpha -1,\beta }^{{C}_{N}}{P}_{\alpha ,\alpha -1}{R}_{\alpha }.$ (C.4)

${\mathsc{𝒮}}_{\alpha -1,\beta }^{{C}_{N}}$ gives the total transition probability from $\beta$ to $\alpha -1$, so the corresponding probability for node $\alpha$ is ${\mathsc{𝒮}}_{\alpha -1,\beta }^{{C}_{N}}$ times the branching probability from $\alpha -1$ to $\alpha$, i.e${P}_{\alpha ,\alpha -1}$, times ${R}_{\alpha }$, which accounts for the weight accumulated from all possible paths that leave and return to node $\alpha$ and are restricted to nodes with indexes greater than $\alpha$. We can now replace ${\mathsc{𝒮}}_{\alpha -1,\beta }^{{C}_{N}}$ by ${\mathsc{𝒮}}_{\alpha -2,\beta }^{{C}_{N}}{P}_{\alpha -1,\alpha -2}{R}_{\alpha -1}$ and so on, until ${\mathsc{𝒮}}_{\alpha ,\beta }^{{C}_{N}}$ is expressed in terms of ${\mathsc{𝒮}}_{\beta ,\beta }^{{C}_{N}}$. Similarly, if $\alpha <\beta$ we have

 ${\mathsc{𝒮}}_{\alpha ,\beta }^{{C}_{N}}={\mathsc{𝒮}}_{\alpha +1,\beta }^{{C}_{N}}{P}_{\alpha ,\alpha +1}{L}_{\alpha },$ (C.5)

and hence

 (C.6)