Markov Chain Monte Carlo (MCMC) methods often take many iterations to converge for highly correlated or high-dimensional target density functions. Methods such as Hamiltonian Monte Carlo (HMC) or No-U-Turn Sampling (NUTS) use the first-order derivative of the density function to tackle the aforementioned issues. However, the calculation of the derivative represents a bottleneck for computationally expensive models. We propose to first build a multi-fidelity Gaussian Process (GP) surrogate. The building block of the multi-fidelity surrogate is a hierarchy of models of decreasing approximation error and increasing computational cost. Then the generated multi-fidelity surrogate is used to approximate the derivative. The majority of the computation is assigned to the cheap models thereby reducing the overall computational cost. The derivative of the multi-fidelity method is used to explore the target density function and generate proposals. We select or reject the proposals using the Metropolis Hasting criterion using the highest fidelity model which ensures that the proposed method is ergodic with respect to the highest fidelity density function. We apply the proposed method to three test cases including some well-known benchmarks to compare it with existing methods and show that multi-fidelity No-U-turn sampling outperforms other methods.

翻译：马尔可夫链蒙特卡洛（MCMC）方法在处理高度相关或高维目标密度函数时，往往需要大量迭代才能收敛。哈密顿蒙特卡洛（HMC）或无回旋采样（NUTS）等方法利用密度函数的一阶导数来解决上述问题。然而，对于计算成本高昂的模型，导数的计算成为一个瓶颈。我们提出首先构建一个多保真度高斯过程（GP）替代模型。该多保真度替代模型的基础是一组近似误差递减而计算成本递增的模型层级。随后，利用生成的多保真度替代模型来近似导数。大部分计算任务分配给廉价模型，从而降低总体计算成本。采用多保真度方法的导数来探索目标密度函数并生成提议点。我们使用最高保真度模型的Metropolis-Hastings准则对提议点进行选择或拒绝，确保所提方法关于最高保真度密度函数具有遍历性。我们将此方法应用于三个测试案例（包括若干知名基准测试），与现有方法进行比较，结果表明多保真度无回旋采样的性能优于其他方法。