Adaptive Belief Propagation – Supplementary Material

Proof. Base case: The messages in the path w1 → w2 are correct. This is trivially true since all the incoming messages to w1 and to the nodes in the path M(w1 → w2) have been correctly evaluated during initialization. Therefore, after we absorb the measurement in the potential of Xw1 , propagating from w1 → w2 will give us the correct messages. Induction step: We will assume now that the messages in M(wj−1 → wj), j ∈ {2, . . . , ` − 1} are correct and we will show that the messages in M(w`−1 → w`) will be correct as well. W.l.o.g. assume the tree is rooted at w`−1 as shown in Fig. 1 and i is one of w`−1’s neighbors. We need to show that all the incoming messages to w`−1 as well as the incoming messages to the other nodes in M(w`−1 → w`) are correct. Let’s first show that the incoming messages to w`−1 are correct. There are three cases for the subtree Ti rooted at i (if we ignore the branch containing the edge (i, w`−1)): (a) there are no previous measurements {w1, . . . , w`−2} from it, (b) the last measurement from it was taken at time ti < ` − 2, or (c) at time ti = ` − 2. In the first case (a), since there are no previous measurements, the incoming message mi→w`−1 stayed intact since initialization and thus is correct. In the second case (b), since ti < ` − 2, this means that at point ti + 1, we moved to a subtree of another neighbor of w`−1 through w`−1. Due to our assumption, that all messages from previous paths M(wj−1 → wj), j < `, are correct, this also implies that the messages in the path M(wti → wti+1) are correct and this includes message mi→w`−1 as well. Lastly, if ti = ` − 2, this means that the previous measurement, at time ` − 2, was taken from the subtree rooted at i (c). By assumption, all messages in M(w`−2 → w`−1) are correct. So, in all cases, the incoming message from i to w`−1 is correct. We follow similar logic for all neighbors of w`−1. Lastly, we should demonstrate that the incoming messages to the other nodes in the pathM(w`−1 → w`) are correct. The logic is similar as before. Let’s refer to the subtrees that are attached Ti