Isothermal titration calorimetry (ITC) is the only experimental technique able to reliably measure both the free energies of protein-ligand binding as well as its decomposition into enthalpic and entropic contributions from a single experiment. Due to the way in which these values are extracted from the raw data, the errors in the free energy, enthalpy, and entropy can be of different magnitudes, and the errors in these quantities highly correlated. Additionally, the use of multiple measurements to improve statistics, competition of weaker ligands to measure the affinities of stronger ligands, and measurements in multiple buffers to separate out the effect of proton uptake can complicate the propagation of these uncertainties to the physical quantities of interest. We present a Bayesian formalism for expressing the full posterior distribution of free energy, entropy, and enthalpy from one or more ITC experiments, allowing their uncertainties and correlations to be quantitatively assessed.