We develop a hierarchical Bayesian inference method to estimate the thermal resistance and the volumetric heat capacity of a wall. These thermal properties are essential for accurate building energy simulations that are needed to make effective energy-saving policies. We apply our methodology to an experimental case study conducted in an environmental chamber, where measurements are recorded every minute from temperature probes and heat flux sensors placed on both sides of a solid brick wall over a five-day period. As a result of smoothing the measurements by local averaging procedure, we can reasonably assume that the temperature and the heat flux measurements have independent Gaussian noise. We model the heat transfer through the wall using the one-dimensional heat equation with unknown Dirichlet boundary conditions. We derive the joint likelihood of the wall parameters and the initial/boundary conditions. We then marginalize the nuisance boundary parameters from the joint likelihood. We obtain approximated Gaussian posterior distributions for the wall parameters and the initial temperature parameter. The results show that our technique reduces the bias error of the estimated wall parameters when compared to other approaches. Finally, we estimate the information gain under two experimental setups to recommend how the user can efficiently determine the duration of the measurement campaign and the range of the external temperature oscillation.