Neal's funnel refers to an exponential tapering in probability densities common to Bayesian hierarchical models. Usual sampling methods, such as Markov Chain Monte Carlo, struggle to efficiently sample the funnel. Reparameterizing the model or analytically marginalizing local parameters are common techniques to remedy sampling pathologies in distributions exhibiting Neal's funnel. In this paper, we show that the challenges of Neal's funnel can be avoided by performing the hierarchical analysis, well, hierarchically. That is, instead of sampling all parameters of the hierarchical model jointly, we break the sampling into multiple stages. The first stage samples a generalized (higher-dimensional) hierarchical model which is parameterized to lessen the sharpness of the funnel. The next stage samples from the estimated density of the first stage, but under a constraint which restricts the sampling to recover the marginal distributions on the hyper-parameters of the original (lower-dimensional) hierarchical model. A normalizing flow can be used to represent the distribution from the first stage, such that it can easily be sampled from for the second stage of the analysis. This technique is useful when effective reparameterizations are computationally expensive to calculate, or a generalized hierarchical model already exists from which it is easy to sample.

翻译：尼尔漏斗指贝叶斯层次模型中常见的概率密度指数锥化现象。常规采样方法（如马尔可夫链蒙特卡洛）难以对漏斗区域进行高效采样。重参数化模型或解析边缘化局部参数是改善呈现尼尔漏斗特征的分布中采样病理的常用技术。本文证明，通过严格按层次结构执行层次分析，可规避尼尔漏斗的采样难题。具体而言，我们不采用联合采样层次模型全部参数的方式，而是将采样过程分解为多个阶段：第一阶段对广义（高维）层次模型进行采样，该模型通过参数化设置缓解漏斗的尖锐程度；第二阶段在约束条件下从第一阶段估计的密度中采样，该约束确保采样结果能恢复原始（低维）层次模型中超参数的边缘分布。规范化流可用于表征第一阶段的分布，从而便于在分析的第二阶段进行采样。当有效重参数化计算成本过高，或已存在易于采样的广义层次模型时，本技术具有重要应用价值。