Estimating True Beliefs from Declared Opinions

Social networks often exert social pressure, causing individuals to adapt their expressed opinions to conform to their peers. An agent in such systems can be modeled as having a (true and unchanging) inherent belief but broadcasts a declared opinion at each time step based on her inherent belief and the past declared opinions of her neighbors. An important question in this setting is parameter estimation: how to disentangle the effects of social pressure to estimate inherent beliefs from declared opinions. To address this, Jadbabaie et al. formulated the interacting P\'olya urn model of opinion dynamics under social pressure and studied it on complete-graph social networks using an aggregate estimator, and found that their estimator converges to the inherent beliefs unless majority pressure pushes the network to consensus. In this work, we study this model on arbitrary networks, providing an estimator which converges to the inherent beliefs even in consensus situations. Finally, we bound the convergence rate of our estimator in both consensus and non-consensus scenarios; to get the bound for consensus scenarios (which converge slower than non-consensus) we additionally found how quickly the system converges to consensus.