Bias and Identifiability in the Bounded Confidence Model

TL;DR

This paper analyzes the statistical properties of maximum likelihood estimators for key parameters in bounded confidence opinion models, highlighting biases and identifiability issues affecting model calibration.

Contribution

It provides a detailed analysis of the biases and identifiability challenges in estimating BCM parameters using maximum likelihood methods.

Findings

Confidence bound estimator is consistent but biased in small samples.

Convergence rate estimator exhibits persistent bias.

Likelihood function analysis reveals multiple local maxima affecting joint parameter estimation.

Abstract

Opinion dynamics models such as the bounded confidence models (BCMs) describe how a population can reach consensus, fragmentation, or polarization, depending on a few parameters. Connecting such models to real-world data could help understanding such phenomena, testing model assumptions. To this end, estimation of model parameters is a key aspect, and maximum likelihood estimation provides a principled way to tackle it. Here, our goal is to outline the properties of statistical estimators of the two key BCM parameters: the confidence bound and the convergence rate. We find that their maximum likelihood estimators present different characteristics: the one for the confidence bound presents a small-sample bias but is consistent, while the estimator of the convergence rate shows a persistent bias. Moreover, the joint parameter estimation is affected by identifiability issues for specific regions of the parameter space, as several local maxima are present in the likelihood function. Our results show how the analysis of the likelihood function is a fruitful approach for better understanding the pitfalls and possibilities of estimating the parameters of opinion dynamics models, and more in general, agent-based models, and for offering formal guarantees for their calibration.