We present a parametrized analytic statistical model of the thermodynamics of alchemical molecular binding within the solvent potential of mean force formalism. The model describes the free energy profiles of linear single-decoupling alchemical binding free energy calculations accurately. The parameters of the model, which are physically motivated, are derived by maximum likelihood inference from data obtained from alchemical molecular simulations. The validity of the model has been assessed on a set of host-guest complexes. The model faithfully reproduces both the binding free energy profiles and the probability densities of the perturbation energy as a function of the alchemical progress parameter. The model offers a rationalization for the characteristic shape of binding free energy profiles. The parameters obtained from the model are potentially useful descriptors of the association equilibrium of molecular complexes. Potential applications of the model for the classification of molecular complexes and the design of alchemical molecular simulations are envisioned.