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.

翻译：我们提出了一种新方法，利用深度学习提升重要性嵌套采样（INS）技术在贝叶斯后验估计与证据计算中的效率。与基于拒绝的采样方法（如标准嵌套采样NS或马尔可夫链蒙特卡洛MCMC算法）不同，重要性采样技术可利用所有似然评估进行后验与证据估计。然而，高效的重要性采样需要能紧密近似后验分布的提议分布。我们展示了如何通过神经网络回归将INS与深度学习结合，以完成这一任务。我们还介绍了NAUTILUS——一个实现该技术的开源Python参考软件包，用于贝叶斯后验估计与证据计算。在多种具有挑战性的合成问题以及系外行星探测、星系SED拟合和宇宙学等实际应用中，我们将NAUTILUS与EMCEE、DYNESTY、ULTRANEST和POCOMC等主流NS与MCMC软件包进行了对比。在所有应用中，NAUTILUS的采样效率均显著高于其他采样器，通常高出一个数量级以上。同时，NAUTILUS能提供高精度结果，且所需似然评估次数少于所有其他测试采样器。我们还证明，NAUTILUS具有良好的似然维度可扩展性，并易于在多CPU环境下并行化。