  We introduce a novel approach to boost the efficiency of the importance
nested sampling (INS) technique for Bayesian posterior and evidence estimation
using deep learning. Unlike rejection-based sampling methods such as vanilla
nested sampling (NS) or Markov chain Monte Carlo (MCMC) algorithms, importance
sampling techniques can use all likelihood evaluations for posterior and
evidence estimation. However, for efficient importance sampling, one needs
proposal distributions that closely mimic the posterior distributions. We show
how to combine INS with deep learning via neural network regression to
accomplish this task. We also introduce NAUTILUS, a reference open-source
Python implementation of this technique for Bayesian posterior and evidence
estimation. We compare NAUTILUS against popular NS and MCMC packages, including
EMCEE, DYNESTY, ULTRANEST and POCOMC, on a variety of challenging synthetic
problems and real-world applications in exoplanet detection, galaxy SED fitting
and cosmology. In all applications, the sampling efficiency of NAUTILUS is
substantially higher than that of all other samplers, often by more than an
order of magnitude. Simultaneously, NAUTILUS delivers highly accurate results
and needs fewer likelihood evaluations than all other samplers tested. We also
show that NAUTILUS has good scaling with the dimensionality of the likelihood
and is easily parallelizable to many CPUs.
