Deep  neural  network  (DNN)  regression  models  are  widely  used  in  applications  requiringstate-of-the-art predictive accuracy.  However,  until recently there has been little work onaccurate uncertainty quantification for predictions from such models.  We add to this liter-ature by outlining an approach to constructing predictive distributions that are ‘marginallycalibrated’.  This is where the long run average of the predictive distributions of the responsevariable  matches  the  observed  empirical  margin.   Our  approach  considers  a  DNN  regres-sion with a conditionally Gaussian prior for the final layer weights, from which an implicitcopula process on the feature space is extracted.  This copula process is combined with anon-parametrically  estimated  marginal  distribution  for  the  response.   The  end  result  is  ascalable distributional DNN regression method with marginally calibrated predictions, andour work complements existing methods for probability calibration.  The approach is firstillustrated  using  two  applications  of  dense  layer  feed-forward  neural  networks.   However,our main motivating applications are in likelihood-free inference, where distributional deepregression is used to estimate marginal posterior distributions.  In two complex ecologicaltime series examples we employ the implicit copulas of convolutional networks,  and showthat marginal calibration results in improved uncertainty quantification.  Our approach alsoavoids the need for manual specification of summary statistics, a requirement that is bur-densome for users and typical of competing likelihood-free inference methods.  This is jointwork with Nadja Klein and Michael Smith.

[1]  N. Klein, D. J. Nott, M.S. Smith. Marginally-calibrated deep distributional regression,J. Comput. Graph. Stat., arXiv:2006.14126, doi 10.1080/10618600.2020.1807996.