  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.
