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.

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