The paper addresses the classical network tomography problem of inferring local traffic given origin-destination observations. Focussing on large complex public transportation systems, we build a scalable model that exploits input-output information to estimate the unobserved link/station loads and the users path preferences. Based on the reconstruction of the users' travel time distribution, the model is flexible enough to capture possible different path-choice strategies and correlations between users travelling on similar paths at similar times. The corresponding likelihood function is intractable for medium or large-scale networks and we propose two distinct strategies, namely the exact maximum-likelihood inference of an approximate but tractable model and the variational inference of the original intractable model. As an application of our approach, we consider the emblematic case of the London Underground network, where a tap-in/tap-out system tracks the start/exit time and location of all journeys in a day. A set of synthetic simulations and real data provided by Transport For London are used to validate and test the model on the predictions of observable and unobservable quantities.

I thank the authors for the clarification in their rebuttal. It is even more clear that the authors should better contrast their work with aggregate approaches such as Dan Sheldon's collective graphical models (e.g., Sheldon and Dietterich (2011), Kumar et al. 2013, Bernstein and Sheldon 2016). Part of the confusion came from some of the modeling choices: In equation (1) the travel times added by one station is Poisson distributed?! Poisson is often used for link loads (how many people there are in a given station), not to model time. Is the quantization of time too coarse for a continuous-time model? Wouldn't a phase-type distribution(e.g., Erlang) be a better choice for time? Such modeling choices must be explained.

The paper addresses the classical network tomography problem of inferring local traffic given origin-destination observations. Focussing on large complex public transportation systems, we build a scalable model that exploits input-output information to estimate the unobserved link/station loads and the users path preferences. Based on the reconstruction of the users' travel time distribution, the model is flexible enough to capture possible different path-choice strategies and correlations between users travelling on similar paths at similar times. The corresponding likelihood function is intractable for medium or large-scale networks and we propose two distinct strategies, namely the exact maximum-likelihood inference of an approximate but tractable model and the variational inference of the original intractable model. As an application of our approach, we consider the emblematic case of the London Underground network, where a tap-in/tap-out system tracks the start/exit time and location of all journeys in a day.