“…This involves replacing h by h − φ in an ergodic average, where φ ∈ L 1 0 (π), so that π(h − φ) = π(h) and the limit of the MCMC estimator is unchanged, but the variance may be smaller with a judicious choice of φ. A convenient family of φ is {(I − P )f : f ∈ L 1 (π)} since φ ∈ L 1 0 (π) by construction and indeed an optimal choice of f is g. Approximations of g have been considered for this purpose in, e.g., Andradóttir et al (1993), Henderson (1997), Dellaportas & Kontoyiannis (2012), Mijatović & Vogrinc (2018), Alexopoulos et al (2022).…”