Iterative maximum likelihood on networks

We consider n agents located on the vertices of a connected graph. Each agent v receives a signal X_v(0) ~ N(¿, 1) where ¿ is an unknown quantity. A natural iterative way of estimating ¿ is to perform the following procedure. At iteration t + 1 let X_v(t + 1) be the average of X_v(t) and of X_w(t) among all the neighbors w of v. It is well known that this procedure converges to X(¿) = 1/2|E|^-1 ¿ d_vX_v where d_v is the degree of v. In this paper we consider a variant of simple iterative averaging, which models ¿greedy¿ behavior of the agents. At iteration t, each agent v declares the value of its estimator X_v(t) to all of its neighbors. Then, it updates X_v(t + 1) by taking the maximum likelihood (or minimum variance) estimator of ¿, given X_v(t) and X_w(t) for all neighbors w of v, and the structure of the graph. We give an explicit efficient procedure for calculating X_v(t), study the convergence of the process as t ¿ ¿ and show that if the limit exists then X_v(¿) = X_w(¿) for all v and w. For graphs that are symmetric under actions of transitive groups, we show that the process is efficient. Finally, we show that the greedy process is in some cases more efficient than simple averaging, while in other cases the converse is true, so that, in this model, ¿greed¿ of the individual agents may or may not have an adverse affect on the outcome. The model discussed here may be viewed as the Maximum-Likelihood version of models studied in Bayesian Economics. The ML variant is more accessible and allows in particular to show the significance of symmetry in the efficiency of estimators using networks of agents.