Slow Convergence of Interacting Kalman Filters in Word-of-Mouth Social Learning

We consider word-of-mouth social learning involving m Kalman filter agents that operate sequentially. The first Kalman filter receives the raw observations, while each subsequent Kalman filter receives a noisy measurement of the conditional mean of the previous Kalman filter. The prior is updated by the m-th Kalman filter. When m=2, and the observations are noisy measurements of a Gaussian random variable, the covariance goes to zero as k-1/3 for k observations, instead of O(k-1) in the standard Kalman filter. In this paper we prove that for m agents, the covariance decreases to zero as k-(2m-1), i.e, the learning slows down exponentially with the number of agents. We also show that by artificially weighing the prior at each time, the learning rate can be made optimal as k-1. The implication is that in word-of-mouth social learning, artificially re-weighing the prior can yield the optimal learning rate.