The typical approach to scattering amplitudes in lattice QCD is based on the finite-volume effects of multi-hadron states. There is however, no direct application of this formalism to inclusive decays where all possible hadronic final states are summed over. An alternative (infinite volume approach) requires the treatment of an inverse problem to obtain inclusive rates directly from Euclidean lattice QCD correlation functions. This framework is discussed first in the two-dimensional O(3) sigma model, where it reproduces analytic results from integrability. An ongoing application to QCD is also discussed.