PATHWAY SUMS FOR CHAIN GRAPHS,

(C.1) |

Similarly, we can continue summing contributions in this way until we have recrossings of to , for which the result of the nested summations is . Hence, is the total transition probability for pathways that return to node and are confined to nodes with index greater than without escape from .

We can similarly calculate the total probability for pathways returning to and confined to nodes with indices smaller than . The total probability factor for recrossings between nodes 1 and 2 is . Hence, the required probability for recrossings between nodes 2 and 3 including arbitrary recrossings between 1 and 2 is . Continuing up to recrossings between nodes and we obtain the total return probability for pathways restricted to this side of as . The general recursive definitions of and are:

We can now calculate as

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

We can now derive as follows. If we can write

(C.4) |

gives the total transition probability from to , so the corresponding probability for node is times the branching probability from to , i.e. , times , which accounts for the weight accumulated from all possible paths that leave and return to node and are restricted to nodes with indexes greater than . We can now replace by and so on, until is expressed in terms of . Similarly, if we have

(C.5) |

and hence