We examine the problem of deflagration of a flammable gaseous mixture contained within an elastic spherical shell and ignited at its centre. We predict analytically the pressure wave radiated by an expanding spherical deflagration front and we study its interaction with the elastic shell prior to failure. We predict the waves reflected at the inner wall of the shell and radiated in the space outside the shell, which we assume initially filled with air at atmospheric pressure, and we calculate the magnitude of these waves as a function of density, thickness and elastic modulus of the shell. The findings of this model are used to critically assess experimental setups used in controlled deflagration experiments, in which a flammable mixture is contained within a soft shell. The interpretation of data from these tests currently assumes that the gaseous mixture is effectively unconfined during the deflagration, due to the low mass, low stiffness and early failure of the containing shell. We show that this assumption is not adequate and it leads to errors in the interpretation of the measurements; the proposed model is used to quantify these errors.