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

arXiv: https://arxiv.org/abs/2410.08447 · HTML: https://mlsanity.com/paper/2410.08447/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 13 references that are in the map.
- First posted on arXiv 11 Oct 2024.

## Builds on (most cited first, 2 of 2)

- [How to Calibrate your Adversary's Capabilities? Inverse Filtering for Counter-Autonomous Systems](https://mlsanity.com/paper/1905.07230/) (2019)
- [Online Reputation and Polling Systems: Data Incest, Social Learning and Revealed Preferences](https://mlsanity.com/paper/1501.00994/) (2015)

## Built on by (most cited first, 1 of 1)

- [Interacting Large Language Model Agents. Interpretable Models and Social Learning](https://mlsanity.com/paper/2411.01271/) (2024)

## Abstract (arXiv)

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.