  We propose a novel approach for solving inverse-problems with
high-dimensional inputs and an expensive forward mapping. It leverages joint
deep generative modelling to transfer the original problem spaces to a lower
dimensional latent space. By jointly modelling input and output variables and
endowing the latent with a prior distribution, the fitted probabilistic model
indirectly gives access to the approximate conditional distributions of
interest. Since model error and observational noise with unknown distributions
are common in practice, we resort to likelihood-free inference with Approximate
Bayesian Computation (ABC). Our method calls on ABC by Subset Simulation to
explore the regions of the latent space with dissimilarities between generated
and observed outputs below prescribed thresholds. We diagnose the diversity of
approximate posterior solutions by monitoring the probability content of these
regions as a function of the threshold. We further analyze the curvature of the
resulting diagnostic curve to propose an adequate ABC threshold. When applied
to a cross-borehole tomography example from geophysics, our approach delivers
promising performance without using prior knowledge of the forward nor of the
noise distribution.
