**A**fter reading the arXiv paper by Korattikara, Chen and Welling, I wondered about the expression of the acceptance step of the Metropolis-Hastings algorithm as a mean of log-likelihoods over the sample. More specifically the long sleepless nights at the hospital led me to ponder the rather silly question of the impact of replacing mean by median. I thus tried running a Metropolis-Hastings algorithm with the substitute and it (of course!) let to a nonsensical answer, as shown by the above graph. The true posterior is the one for a normal model and the histogram indicates a lack of convergence of the Markov chain to this posterior even though it does converge to some posterior. Here is the R code for this tiny experiment:

#data generation N=100 x=rnorm(N) #HM steps T=10^5 theta=rep(0,T) curlike=dnorm(x,log=TRUE) for (t in 2:T){ prop=theta[t-1]+.1*rnorm(1) proplike=dnorm(x,mean=prop,log=TRUE) u=runif(1) bound=log(u)-dnorm(prop,sd=10,log=TRUE)+ dnorm(theta[t-1],sd=10,log=TRUE) if (median(proplike)-median(curlike)>bound/N){ theta[t]=prop;curlike=proplike } else { theta[t]=theta[t-1]} }