The Limits to Learning a Diffusion Model

TL;DR

This paper establishes fundamental lower bounds on the sample complexity for estimating diffusion models like Bass and SIR, showing that accurate predictions require observing the system well past the peak of the diffusion process.

Contribution

It provides the first theoretical lower bounds on the sample complexity for learning simple diffusion models, highlighting the difficulty of early prediction.

Findings

Estimation of diffusion models is fundamentally limited until late in the process.

Early prediction of the final adoption or infection count is impossible before two-thirds of the peak.

Additional data sources, like seroprevalence studies, can improve model estimation.

Abstract

This paper provides the first sample complexity lower bounds for the estimation of simple diffusion models, including the Bass model (used in modeling consumer adoption) and the SIR model (used in modeling epidemics). We show that one cannot hope to learn such models until quite late in the diffusion. Specifically, we show that the time required to collect a number of observations that exceeds our sample complexity lower bounds is large. For Bass models with low innovation rates, our results imply that one cannot hope to predict the eventual number of adopting customers until one is at least two-thirds of the way to the time at which the rate of new adopters is at its peak. In a similar vein, our results imply that in the case of an SIR model, one cannot hope to predict the eventual number of infections until one is approximately two-thirds of the way to the time at which the infection rate has peaked. This lower bound in estimation further translates into a lower bound in regret for decision-making in epidemic interventions. Our results formalize the challenge of accurate forecasting and highlight the importance of incorporating additional data sources. To this end, we analyze the benefit of a seroprevalence study in an epidemic, where we characterize the size of the study needed to improve SIR model estimation. Extensive empirical analyses on product adoption and epidemic data support our theoretical findings.