C Pathway Sums for Chain Graphs, 𝒮CN α,β

Appendix C
Pathway Sums for Chain Graphs, $𝒮_{α, β}^{C_{N}}$

To obtain the total probability of leaving the chain $C_{N}$ via node $α$ if started from node $β$ , i.e. $ℰ_{α}^{C_{N}} 𝒮_{α, β}^{C_{N}}$ , we must calculate the pathway sum $𝒮_{α, β}^{C_{N}}$ . We start with the case $α = β$ and obtain $𝒮_{β, β}^{C_{N}}$ . Consider any path that has reached node $N - 1$ . The probability factor due to all possible $N - 1$ to $N$ recrossings is simply $R_{N - 1} = 1 ∕ (1 - P_{N - 1, N} P_{N, N - 1})$ . 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} {(P_{N - 2, N - 1} P_{N - 1, N - 2} R_{N - 1})}^{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 $β$ to $β + 1$ , for which the result of the nested summations is $R_{β} = 1 ∕ (1 - P_{β, β + 1} P_{β + 1, β} R_{β + 1})$ . Hence, $R_{β}$ is the total transition probability for pathways that return to node $β$ and are confined to nodes with index greater than $β$ without escape from $C_{N}$ .

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

\begin{matrix} L_{j} & = & \begin{matrix} 1, & j = 1, \\ 1 ∕ (1 - P_{j - 1, j} P_{j, j - 1} L_{j - 1}), & j > 1, \end{matrix} \\ and R_{j} & = & \begin{matrix} 1, & j = N, \\ 1 ∕ (1 - P_{j, j + 1} P_{j + 1, j} R_{j + 1}), & j < N . \end{matrix} \end{matrix}

(C.2)

We can now calculate $𝒮_{β, β}^{C_{N}}$ as

\begin{matrix} 𝒮_{β, β}^{C_{N}} & = & \sum_{m = 0}^{\infty} \sum_{n = 0}^{m} \frac{n!}{m! (n - m)!} {P_{β - 1, β} P_{β, β - 1} L_{β - 1})}^{n} {P_{β, β + 1} P_{β + 1, β} R_{β + 1})}^{m - n} \\ = & \sum_{m = 0}^{\infty} {(P_{β - 1, β} P_{β, β - 1} L_{β - 1} + P_{β, β + 1} P_{β + 1, β} R_{β + 1})}^{m} \\ = & {(1 - P_{β - 1, β} P_{β, β - 1} L_{β - 1} - P_{β, β + 1} P_{β + 1, β} R_{β + 1})}^{- 1} \\ = & {1 - \frac{L_{β} - 1}{L_{β}} - \frac{R_{β} - 1}{R_{β}})}^{- 1} \\ = & \frac{L_{β} R_{β}}{L_{β} - L_{β} R_{β} + R_{β}}, \end{matrix}

(C.3)

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

We can now derive $𝒮_{α, β}^{C_{N}}$ as follows. If $α > β$ we can write

𝒮_{α, β}^{C_{N}} = 𝒮_{α - 1, β}^{C_{N}} P_{α, α - 1} R_{α} .

(C.4)

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

𝒮_{α, β}^{C_{N}} = 𝒮_{α + 1, β}^{C_{N}} P_{α, α + 1} L_{α},

(C.5)

and hence

𝒮_{α, β}^{C_{N}} = \begin{matrix} 𝒮_{β, β}^{C_{N}} \prod_{i = α}^{β - 1} P_{i, i + 1} L_{i}, & α < β, \\ 𝒮_{β, β}^{C_{N}} \prod_{i = β + 1}^{α} P_{i, i - 1} R_{i}, & α > β . \end{matrix}

(C.6)

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Appendix CPathway Sums for Chain Graphs, 𝒮α,βCN

Appendix C
Pathway Sums for Chain Graphs, $𝒮_{α, β}^{C_{N}}$