  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.
