  The eigenstate thermalisation hypothesis (ETH) is a statistical
characterisation of eigen-energies, eigenstates and matrix elements of local
operators in thermalising quantum systems. We develop an ETH-like ansatz of a
partially thermalising system composed of a spin-1/2 coupled to a finite
quantum bath. The spin-bath coupling is sufficiently weak that ETH does not
apply, but sufficiently strong that perturbation theory fails. We calculate (i)
the distribution of fidelity susceptibilities, which takes a broadly
distributed form, (ii) the distribution of spin eigenstate entropies, which
takes a bi-modal form, (iii) infinite time memory of spin observables, (iv) the
distribution of matrix elements of local operators on the bath, which is
non-Gaussian, and (v) the intermediate entropic enhancement of the bath, which
interpolates smoothly between zero and the ETH value of $\log 2$. The
enhancement is a consequence of rare many-body resonances, and is
asymptotically larger than the typical eigenstate entanglement entropy. We
verify these results numerically and discuss their connections to the many-body
localisation transition.
