Iterative Maximum Likelihood on Networks
leaning on networks
algebraic recursion relation
Statistics and Probability
We consider n agents located on the vertices of a connected graph. Each agent v receives a signal Xv(0)∼N(μ,1) where μ is an unknown quantity. A natural iterative way of estimating μ is to perform the following procedure. At iteration t+1 let Xv(t+1) be the average of Xv(t) and of Xw(t) among all the neighbors w of v. It is well known that this procedure converges to X (∞) = ½ |E|-1 ∑dvXv where dv 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 Xv(t) to all of its neighbors. Then, it updates Xv(t+1) by taking the maximum likelihood (or minimum variance) estimator of μ, given Xv(t) and Xw(t) for all neighbors w of v, and the structure of the graph. We give an explicit efficient procedure for calculating Xv(t), study the convergence of the process as t→∞ and show that if the limit exists then Xv(∞)=Xw(∞) 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.