  Estimating the duration of user behavior is a central concern for most
internet companies. Survival analysis is a promising method for analyzing the
expected duration of events and usually assumes the same survival function for
all subjects and the event will occur in the long run. However, such
assumptions are inappropriate when the users behave differently or some events
never occur for some users, i.e., the conversion period on web services of the
light users with no intention of behaving actively on the service. Especially,
if the proportion of inactive users is high, this assumption can lead to
undesirable results. To address these challenges, this paper proposes a mixture
model that separately addresses active and inactive individuals with a latent
variable. First, we define this specific problem setting and show the
limitations of conventional survival analysis in addressing this problem. We
demonstrate how naturally our Bernoulli-Weibull model can accommodate the
challenge. The proposed model was extended further to a Bayesian hierarchical
model to incorporate each subject's parameter, offering substantial
improvements over conventional, non-hierarchical models in terms of WAIC and
WBIC. Second, an experiment and extensive analysis were conducted using
real-world data from the Japanese job search website, CareerTrek, offered by
BizReach, Inc. In the analysis, some research questions are raised, such as the
difference in activation rate and conversion rate between user categories, and
how instantaneously the rate of event occurrence changes as time passes.
Quantitative answers and interpretations are assigned to them. Furthermore, the
model is inferred in a Bayesian manner, which enables us to represent the
uncertainty with a credible interval of the parameters and predictive
quantities.
