Non-Bayesian Social Learning with Gaussian Uncertain Models

Abstract

Non-Bayesian social learning theory provides a framework for distributed inference of a group of agents interacting over a social network by sequentially communicating and updating beliefs about the unknown state of the world through likelihood updates from their observations. Typically, likelihood models are assumed known precisely. However, in many situations the models are generated from sparse training data due to lack of data availability, high cost of collection/calibration, limits within the communications network, and/or the high dynamics of the operational environment. Recently, social learning theory was extended to handle those model uncertainties for categorical models. In this paper, we introduce the theory of Gaussian uncertain models and study the properties of the beliefs generated by the network of agents. We show that even with finite amounts of training data, non-Bayesian social learning can be achieved and all agents in the network will converge to a consensus belief that provably identifies the best estimate for the state of the world given the set of prior information.This research was sponsored by the DARPA Lagrange, Vannevar Bush Fellowship, and OSD LUCI programs.J.Z.H. and L.K.

A General Framework for Distributed Inference with Uncertain Models

“…Then, Corollary III.2.1 shows that as the amount of prior evidence grows without bound, only the hypothesis with parameters 𝜑 𝜃 = 𝜑 𝜃 * = arg min 𝜑∈Φ 𝐷 𝐾𝐿 (𝑄‖𝑃 (•|𝜑)) will have a ratio > 0, allowing the agents to learn the set of hypotheses indistinguishable with the ground truth, i.e., Θ * . These results are consistent with but more general than the uncertain models presented in [30], [32].…”

“…Additionally, we identify conditions that allow the agents to include model uncertainty [34], i.e., the true statistical model is not within the parametric family of distributions. The uncertain models are implemented into a non-Bayesian social learning rule and show that the results are consistent and that the works presented in [30], [32] are special cases of the uncertain models presented herein. Additionally, we provide an algorithmic representation of how to implement uncertain models in a practical setting for continuous and discrete observations.…”

“…In this section, we present a general class of uncertain models and their asymptotic properties for a single agent that has collected a limited amount of training data for each class [30]- [32]. For simplicity and ease of presentation, we drop the subscript 𝑖 in our notation for the remainder of this section.…”

“…Initially, uncertain models were developed to handle categorical observations and prior evidence drawn from multinomial distributions [30] and were later extended for data drawn from univariate Gaussian distributions [32]. These models were implemented into a non-Bayesian social learning rule and studied for static and time-varying graphs [31].…”

“…In this work, we build upon [30], [32] by identifying a broad set of parametric distributions that enable learning with uncertain models. Additionally, we identify conditions that allow the agents to include model uncertainty [34], i.e., the true statistical model is not within the parametric family of distributions.…”