## Statistics Papers

#### Document Type

Journal Article

#### Date of this Version

7-2010

#### Publication Source

Advances in Applied Mathematics

#### Volume

45

#### Issue

1

#### Start Page

36

#### Last Page

49

#### DOI

10.1016/j.aam.2009.11.004

#### Abstract

We consider *n* agents located on the vertices of a connected graph. Each agent *v* receives a signal X_{v}(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 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 (∞) = ½ |E|^{-1} ∑d_{v}X_{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 *μ*, 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.

#### Copyright/Permission Statement

© 2010. This manuscript version is made available under the CC-BY-NC-ND 4.0 license

#### Keywords

maximum likelihood, sparse sensing, leaning on networks, algebraic recursion relation, bayesian economics

#### Recommended Citation

Mossel, E.,
&
Tamuz, O.
(2010).
Iterative Maximum Likelihood on Networks.
*Advances in Applied Mathematics,*
*45*
(1),
36-49.
http://dx.doi.org/10.1016/j.aam.2009.11.004

**Date Posted:** 27 November 2017

This document has been peer reviewed.

## Comments

At the time of publication, author Elchanan Mossel was affiliated with the University of California, Berkeley. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania.