Estimating true beliefs from declared opinions

A common feature of interactions and opinion exchanges on social networks, both real and digital, is the presence of social pressure, which may cause agents to alter their expressed opinions in order to fit in with those around them. In such systems, each agent has a true and unchanging inherent belief but broadcasts a declared opinion at each time step, influenced by both her inherent belief and the declared opinions of her neighbors. An important question in this setting is parameter estimation: how to disentangle the effects of social pressure and estimate the underlying true beliefs of the agents from their declared opinions. To address this question, Jadbabaie et al. [1] formulated the interacting Polya urn model of opinion dynamics under social pressure and studied parameter estimation on complete-graph social networks using an aggregate estimator. They found that, under these settings, this estimator asymptotically estimates the true beliefs unless majority pressure causes the network to approach consensus over time. In this work, we consider parameter estimation for the interacting Polya urn model on arbitrary networks, and prove that the maximum likelihood estimator always asymptotically estimates the true beliefs - including the degree to which those beliefs are held - even when consensus is approached.