The Limits to Learning a Diffusion Model

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 Susceptible-Infected-Recovered (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 the Bass model, our results imply that when new adopters are predominantly driven by imitation, 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. This paper was accepted by David Simchi-Levi, data science. Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 1727239]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.02953 .