## A vanilla Rao-Blackwellisation (2)

Following yesterday’s post on our vanilla Rao-Blackwellisation, Nicolas Chopin pointed out to me today a related paper by Hugo Hammer and Hakon Tjelmeland, Control Variates for the Metropolis–Hastings Algorithm, that was published last year in the Scandinavian Journal of Statistics. The approach they adopt is to introduce control variates based on the Metropolis-Hastings Markov chain as well as on the proposed moves. Those control variates are based on zero-mean functions

$g(x_t,y_t) = w_1(x_t,y_t) f(x_t) +w_2(x_t,y_t) f(y_t)$

with

$w_1(x,y)\pi(x)q(x,y) = -w_2(x,y)\pi(y)q(y,x)$

which leads to both the Barker (1965) and the traditional $\alpha(x,y)$ acceptance probabilities. The main connection with our vanilla Rao-Blackwellisation paper is that the computation burden of using control variates is also of order $\text{O}(N)$, because the variance improvement brought by the control variate technique [as opposed to Rao-Blackwellisation] is difficult to assess theoretically when the optimal regression coefficient in the control variate estimate is approximated by the empirical correlation across MCMC iterations. Further, the control variate improvement is “local” in that it only depends on $(x_t,y_t)$ at iteration t rather than on the sequence of proposed values as in Rao-Blackwellization of sampling schemes or on the sequence of accepted values and rejected proposals as in the vanilla Rao-Blackwellisation paper. An interesting application of the paper deals with reversible jump settings since both the control variate and the vanilla Rao-Blackwellisation techniques apply to functions of the Markov chain that remain defined across models.

Arnaud Guillin also kindly pointed out a typo in the Metropolis-Hastings acceptance probability, typo that now stands corrected on arXiv!