A site's recommendation system relies on knowledge of its users' preferences to offer relevant recommendations to them. These preferences are for attributes that comprise items and content shown on the site, and are estimated from the data of users' interactions with the site. Another form of users' preferences is material too, namely, users' preferences for the site over other sites, since that shows users' base level propensities to engage with the site. Estimating users' preferences for the site, however, faces major obstacles because (a) the focal site usually has no data of its users' interactions with other sites; these interactions are users' unobserved behaviors for the focal site; and (b) the Machine Learning literature in recommendation does not offer a model of this situation. Even if (b) is resolved, the problem in (a) persists since without access to data of its users' interactions with other sites, there is no ground truth for evaluation. Moreover, it is most useful when (c) users' preferences for the site can be estimated at the individual level, since the site can then personalize recommendations to individual users. We offer a method to estimate individual user's preference for a focal site, under this premise. In particular, we compute the focal site's share of a user's online engagements without any data from other sites. We show an evaluation framework for the model using only the focal site's data, allowing the site to test the model. We rely upon a Hierarchical Bayes Method and perform estimation in two different ways - Markov Chain Monte Carlo and Stochastic Gradient with Langevin Dynamics. Our results find good support for the approach to computing personalized share of engagement and for its evaluation.

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