Modeling ferroelectric materials from first principles is one of the successes of density-functional theory and the driver of much development effort, requiring an accurate description of the electronic processes and the thermodynamic equilibrium that drive the spontaneous symmetry breaking and the emergence of macroscopic polarization. We demonstrate the development and application of an integrated machine learning model that describes on the same footing structural, energetic, and functional properties of barium titanate (BaTiO3), a prototypical ferroelectric. The model uses ab initio calculations as a reference and achieves accurate yet inexpensive predictions of energy and polarization on time and length scales that are not accessible to direct ab initio modeling. These predictions allow us to assess the microscopic mechanism of the ferroelectric transition. The presence of an order-disorder transition for the Ti off-centered states is the main driver of the ferroelectric transition, even though the coupling between symmetry breaking and cell distortions determines the presence of intermediate, partly-ordered phases. Moreover, we thoroughly probe the static and dynamical behavior of BaTiO3 across its phase diagram without the need to introduce a coarse-grained description of the ferroelectric transition. Finally, we apply the polarization model to calculate the dielectric response properties of the material in a full ab initio manner, again reproducing the correct qualitative experimental behavior.