We introduce a computational framework to statistically infer thermophysical properties of any given wall from in-situmeasurements of air temperature and surface heat fluxes. The proposed framework uses these measurements, within a Bayesian calibration approach, to sequentially infer input parameters of a one-dimensional heat diffusion model that describes the thermal performance of the wall. These inputs include spatially-variable functions that characterise the thermal conductivity and the volumetric heat capacity of the wall. We encode our computational framework in an algorithm that sequentially updates our probabilistic knowledge of the thermophysical properties as new measurements become available, and thus enables an on-the-fly uncertainty quantification of these properties. In addition, the proposed algorithm enables us to investigate the effect of the discretisation of the underlying heat diffusion model on the accuracy of estimates of thermophysical properties and the corresponding predictive distributions of heat flux. By means of virtual/synthetic and real experiments we show the capabilities of the proposed approach to (i) characterise heterogenous thermophysical properties associated with, for example, unknown cavities and insulators: (ii) obtain rapid and accurate uncertainty estimates of effective thermal properties (e.g. thermal transmittance): and (iii) accurately compute an statistical description of the thermal performance of the wall which is, in turn, crucial in evaluating possible retrofit measures. (C) 2018 Elsevier B.V. All rights reserved.