Following a suggestion by Christian Hennig at JSM 2024, I started working with my PhD student Dana Naderi on a detailed assessment of the method we proposed in 2009 with Darren Wraith for evidence approximation. (The method was briefly mentioned in a Physical Review paper and also briefly illustrated in our 2010 San Antonio survey of evidence approximation methods with Jean-Michel Marin.) Well,  it took longer than expected but we eventually completed our paper on the approximation of evidence by bounded harmonic means, exploiting the general identity of Alan Gelfand and Dipak Dey (1994). Following our 2009 idea, the free function in Gelfand & Dey representation is chosen as a Uniform distribution on an HPD region, since this insures boundedness (and hence finite variance) for the resulting estimator. This followed a revival of the method, renamed THAMES, by Metodiev et al. in 2023, where the authors approximate the HPD region with an ellipsoid derived from a Normal distribution centred at the highest of the HPD points, whose covariance matrix is estimated from the posterior sample. With the drawback that this ellipsoid may as well include low probability regions. Our approach (ECMLE, standing for elliptical coverings for marginal likelihood estimation) is aiming at staying within the actual and targeted HPD region by creating non-overlapping ellipsoids from simulations from the posterior and by using a Uniform density on that collection as the reverse importance function. The resulting estimator is unbiased, since the volume of the set is known (when based on a second, independent, sample of simulations from the posterior, a requirement that was not clearly explicated in our earlier survey). In the meanwhile, that is, while we were close to conclude our paper, Metodiev et al. produced a modified version of THAMES, where they truncate the original ellipsoid to intersect with the HPD region of interest, as discussed in an earlier ‘Og entry (with a reply from the authors). While their paper is more focussed on mixture inference and include other aspects on Bayesian inference for mixture, we rewrote ours to include a comparison between the methods, which proves satisfactory, especially in larger dimensions.