  We present a study on the nonperturbative calculation of observables for
inclusive semileptonic decays of $B_{(s)}$ mesons using lattice QCD. We focus
on the comparison of two different methods to analyse the lattice data of
Euclidean correlation functions, specifically Chebyshev and Backus-Gilbert
approaches. This type of computation may eventually provide new insight into
the long-standing tension between the inclusive and exclusive determinations of
the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements $|V_{cb}|$ and $|V_{ub}|$.
We report the results from a pilot lattice computation for the decay $B_s
\rightarrow X_c \, l\nu_l$, where the valence quark masses are approximately
tuned to their physical values using the relativistic-heavy quark action for
the $b$ quark and the domain-wall formalism for the other valence quarks. We
address the computation of the total decay rate as well as leptonic and
hadronic moments, discussing similarities and differences between the two
analysis techniques.

High Energy Physics - Lattice