Lagrangian stochastic turbulence models offer a uniquely rich and intuitive framework for dealing with turbulence. By directly describing turbulent transport they capture the essential features of turbulence. Stochastic turbulence models are also in an immediate correspondence to the traditional modeling approaches and can be used for deriving realizable RANS and LES closures. At the same time stochastic Lagrangian models are closely related to the methods of modern nonequilibrium statistical mechanics and can draw inspiration from them. Yet the aforementioned virtues of the stochastic Lagrangian framework are underutilized, with no turbulent modeling approaches that would fully use the advantages of the statistical Lagrangian description. We attribute this to the relative difficulty of measuring the parameters (transport coefficients) of stochastic models from experiments and direct numerical simulations (DNS). For example, even for the most promising class of stochastic models, the generalized Langevin model (GLM), there are no methods to measure its parameters, the drift and diffusion tensors, that would work in general nonhomogeneous and nonstationary flows. It is challenging to develop any physical theory without an access to solid experimental data, even more so in the infamously unyieldly case of turbulence. In this work we develop inference methods that can be used to measure the drift and diffusion tensors of the GLM in general flows. These new inference techniques are extensively tested against DNS data for channel flow. Of particular interest are the Green--Kubo methods that originate in nonequilibrium statistical physics. With the newfound ability to measure the parameters of the GLM we investigated the validity of the GLM itself. We established that the GLM is a promising form for turbulence closure that is generally valid throughout the whole channel flow up to the very wall. We also argue how the inference methods based on the Lagrangian acceleration statistics, like the Green--Kubo formulas, can be used to find theoretical closures for the GLM. We show how the models for the drift and diffusion are linked to the small scale structure of turbulence. In the case of homogeneous isotropic turbulence we were able to close the GLM in terms of the dynamics of the velocity gradient tensor, a quantity currently subject to intense research.