A Bayesian data analysis technique is presented as a general tool for inverting linear viscoelastic models of branched polymers. The proposed method takes rheological data of an unknown polymer sample as input and provides a distribution of compositions and structures consistent with the rheology, as its output. It does so by converting the inverse problem of analytical rheology into a sampling problem, using the idea of Bayesian inference. A Markov chain Monte Carlo method with delayed rejection is proposed to sample the resulting posterior distribution. As an example, the method is applied to pure linear and star polymers and linear-linear, star-star, and star-linear blends. It is able to (a) discriminate between pure and blend systems, (b) accurately predict the composition of the mixtures, in the absence of degenerate solutions, and (c) describe multiple solutions, when more than one possible combination of constituents is consistent with the rheology.