Information Theory, Inference, and Learning ... - Inference Group

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Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.242 16 — Message Passingthe number said to him by thesoldier in front of him,(which equals the total number ofsoldiers in front)+ the number said to the commander(which is the number behind)by the soldier behindhim,+ one (to count the commander himself).This solution requires only local communication hardware and simple computations(storage and addition of integers).Separation142332Commander4 Figure 16.2. A line of soldierscounting themselves usingmessage-passing rule-set A. The1commander can add ‘3’ from thesoldier in front, ‘1’ from thesoldier behind, and ‘1’ for himself,and deduce that there are 5soldiers in total.This clever trick makes use of a profound property of the total number ofsoldiers: that it can be written as the sum of the number of soldiers in frontof a point and the number behind that point, two quantities which can becomputed separately, because the two groups are separated by the commander.If the soldiers were not arranged in a line but were travelling in a swarm,then it would not be easy to separate them into two groups in this way. TheFigure 16.3. A swarm of guerillas.CommanderJimguerillas in figure 16.3 could not be counted using the above message-passingrule-set A, because, while the guerillas do have neighbours (shown by lines),it is not clear who is ‘in front’ and who is ‘behind’; furthermore, since thegraph of connections between the guerillas contains cycles, it is not possiblefor a guerilla in a cycle (such as ‘Jim’) to separate the group into two groups,‘those in front’, and ‘those behind’.A swarm of guerillas can be counted by a modified message-passing algorithmif they are arranged in a graph that contains no cycles.Rule-set B is a message-passing algorithm for counting a swarm of guerillaswhose connections form a cycle-free graph, also known as a tree, as illustratedin figure 16.4. Any guerilla can deduce the total in the tree from the messagesthat they receive.
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.16.1: Counting 243Figure 16.4. A swarm of guerillaswhose connections form a tree.CommanderJim1. Count your number of neighbours, N.2. Keep count of the number of messages you have received from yourneighbours, m, and of the values v 1 , v 2 , . . . , v N of each of thosemessages. Let V be the running total of the messages you havereceived.Algorithm 16.5. Message-passingrule-set B.3. If the number of messages you have received, m, is equal to N − 1,then identify the neighbour who has not sent you a message and tellthem the number V + 1.4. If the number of messages you have received is equal to N, then:(a) the number V + 1 is the required total.(b) for each neighbour n {say to neighbour n the number V + 1 − v n .}A Figure 16.6. A triangular 41 × 41grid. How many paths are therefrom A to B? One path is shown.B
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