  We report the application of implicit likelihood inference to the prediction
of the macro-parameters of strong lensing systems with neural networks. This
allows us to perform deep learning analysis of lensing systems within a
well-defined Bayesian statistical framework to explicitly impose desired priors
on lensing variables, to obtain accurate posteriors, and to guarantee
convergence to the optimal posterior in the limit of perfect performance. We
train neural networks to perform a regression task to produce point estimates
of lensing parameters. We then interpret these estimates as compressed
statistics in our inference setup and model their likelihood function using
mixture density networks. We compare our results with those of approximate
Bayesian neural networks, discuss their significance, and point to future
directions. Based on a test set of 100,000 strong lensing simulations, our
amortized model produces accurate posteriors for any arbitrary confidence
interval, with a maximum percentage deviation of $1.4\%$ at $21.8\%$ confidence
level, without the need for any added calibration procedure. In total,
inferring 100,000 different posteriors takes a day on a single GPU, showing
that the method scales well to the thousands of lenses expected to be
discovered by upcoming sky surveys.
