Coupled unbiased MCMC for estimating Shapley effects under dependent inputs

In global sensitivity analysis, Shapley effects provide a game-theoretic allocation of output variance among potentially dependent inputs. Their computation typically requires repeated conditional sampling, which can be challenging even when independent samples from the joint input distribution can be generated exactly. This difficulty arises naturally in hierarchical models, where conditioning on arbitrary subsets need not follow the generative order. To efficiently estimate Shapley effects under such dependence between input variables, we propose a permutation-based framework using coupled unbiased Markov chain Monte Carlo (MCMC). Our proposed method combines two complementary coupling mechanisms: (i) a joint coupling of conditional distributions along a random permutation, designed to reduce the variance of estimated marginal contributions, and (ii) a meeting-based coupling of Markov chains that removes finite-time MCMC bias through a telescoping correction. Under suitable conditions, we prove that the resulting estimators for the unnormalized Shapley effects are unbiased, and have finite variance and finite expected computational cost. Numerical experiments with various test cases illustrate the computational efficiency of the proposed method.