In Stein Π-Importance Sampling, Congye Wang et al. (mostly from Newcastle, UK) build an MCMC scheme with invariant distribution Π targeting a distribution P, showing that the optimal solution (in terms of a discrepancy) differs from P when the chain is Stein-sampled, e..g. via kernel discrepancies. In terms of densities, the solution is
the correction involving the root of a Stein kernel, introduced by Oates, Girolami, and Chopin in their 2017 Series B Read Paper. This is rather paradoxical, even though the outcome does depend on the divergence criterion. Most intriguing!!!
